Geometry Unit 3: Parallel and Perpendicular Relationships Time Frame: Approximately three weeks Unit Description This unit demonstrates the basic role played by Euclid s fifth postulate in geometry. Euclid s fifth postulate is stated in most textbooks using the wording found in Playfair s Axiom: Through a given point, only one line can be drawn parallel to a given line. This axiom and several others are considered by some mathematicians to be equivalent to Euclid s fifth postulate. The focus is on basic angle measurement relationships for parallel and perpendicular lines, equations of lines that are parallel and perpendicular in the coordinate plane, and proving that two or more lines are parallel using various methods including distance between two lines. Student Understandings Students should know the basic angle measurement relationships and slope relationships between parallel and perpendicular lines in the plane. Students can write and identify equations of lines that represent parallel and perpendicular lines. They can recognize the conditions that must exist for two or more lines to be parallel. Three-dimensional figures can be connected to their 2-dimensional counterparts when possible. Guiding Questions. Can students relate parallelism to Euclid s fifth postulate and its ramifications for Euclidean Geometry? 2. Can students use parallelism to find and develop the basic angle measurements related to triangles and to transversals intersecting parallel lines? 3. Can students link perpendicularity to angle measurements and to its relationship with parallelism in the plane and 3-dimensional space? 4. Can students solve problems given the equations of lines that are perpendicular or parallel to a given line in the coordinate plane and discuss the slope relationships governing these situations? 5. Can students solve problems that deal with distance on the number line or in the coordinate plane? 6. Can students solve problems that deal with dividing a directed line segment into a given ratio? Geometry Unit 3 Parallel and Perpendicular Relationships 3-
Unit 3 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) Grade-Level Expectations GLE # GLE Text and Benchmarks Algebra 6. Write the equation of a line parallel or perpendicular to a given line through a specific point (A-3-H) (G-3-H) Geometry 2. Apply the Pythagorean theorem in both abstract and real-life settings (G-2-H) 6. Represent and solve problems involving distance on a number line or in the plane (G-3-H) 9. Develop formal and informal proofs (e.g., Pythagorean theorem, flow charts, paragraphs) (G-6-H) Data Analysis, Probability, and Discrete Math 22. Interpret and summarize a set of experimental data presented in a table, bar graph, line graph, scatter plot, matrix, or circle graph (D-7- H) CCSS for Mathematical Content CCSS # CCSS Text Congruence G.CO. Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Expressing Geometric Properties with Equations G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. ELA CCSS CCSS # CCSS Text Reading Standards for Literacy in Science and Technical Subjects 6-2 RST.9-0.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. RST.9-0.4 Determine the meaning of symbols, key terms, and other domainspecific words and phrases as they are used in a specific scientific or technical context relevant to grades 9-0 texts and topics. Writing Standards for Literacy in History/Social Studies, Science and Technical Subjects 6-2 WHST.9-0.b Write arguments focused on discipline-specific content. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience s knowledge level and concerns. Geometry Unit 3 Parallel and Perpendicular Relationships 3-2
WHST.9-0.2d WHST.9-0.0 Write informative/explanatory texts, including the narration of historical events, scientific procedures/experiments, or technical processes. Use precise language and domain-specific vocabulary to manage the complexity of the topic and convey a style appropriate to the discipline and context as well as to the expertise of likely readers. Write routinely over extended time frames (time for reflection and revision) and shorter time frames (a single sitting or a day or two) for a range of discipline-specific tasks, purposes, and audiences. Sample Activities Activity : Slopes and Equations of Parallel and Perpendicular Lines (GLE: 6, 22; CCSS: G.CO., RST.9-0.4) Materials List: pencil, paper, graphs of lines for work in pairs, graph paper, computer drawing program (optional), vocabulary cards from Unit 2 Begin by having students work in pairs. Provide each pair of students with three graphs of lines, each on a separate coordinate grid system. On one set of axes, have a graph of parallel lines; on the other two sets of axes, have single lines with different slopes and y- intercepts. Be sure to use a different set of lines for each pair of students. Provide at least two points on each line by marking the points on the graphs. Review how to find the slope of a line from two points, and then have students determine the slope of each of the lines provided. Next, have students carefully fold each of the single lines onto itself and crease the paper along the fold line. Have students measure the angle formed by the line and the crease line to confirm that it is a right angle. Be sure to engage students in a discussion that helps them see that the crease line is perpendicular to the original line. Students should develop a convincing argument that the crease line is perpendicular to the original lines (i.e., The two angles are congruent because they are the same size since the angles match when folded over one another. Because a line measures 80 degrees, the measures of the two angles are 90 degrees each). Students will then determine the slope of the crease line and compare it to that of the original line. All data from the class should be recorded in a chart. The chart should include a column for the slope of the original line and a column for the slope of the crease line. Using the class data, student pairs should make a conjecture about the slope of perpendicular lines. Review with students the process for developing the equation of the line if two points on the line are given and have them write the equations of the given lines and the crease line. Students should begin to identify characteristics of equations of perpendicular lines to be developed further in this activity. Geometry Unit 3 Parallel and Perpendicular Relationships 3-3
Using the graph with the given parallel lines, have students fold the graph so each line lies on itself to create a crease line which passes through the pair of parallel lines. Then have students discuss how the slopes of the parallel lines are related, and whether the crease is perpendicular to one or both of the given lines. This should lead to a discussion about the theorem that states, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. Have students write the equations of the given parallel lines and the crease line. Students should again begin to identify characteristics of equations of parallel lines to be developed further in this activity. Provide students with several equations of pairs of lines that are parallel, and several equations of pairs of lines that are perpendicular without giving the relationships. Have students graph the lines and determine the characteristics of the equations of the lines that are parallel, and the characteristics of those that are perpendicular (i.e., parallel lines have the same slopes and different y-intercepts, perpendicular lines have slopes that are opposite reciprocals of one another). As an alternative, have students use a computer software program like Geometer s Sketchpad to draw a pair of perpendicular lines and a pair of parallel lines. Have the program generate the equations of those lines and then determine the characteristics of the equations. Have students refer to the equations they wrote at the beginning of the activity and compare them to the equations they were given to determine if the characteristics they have identified apply to all of the equations they have. Once the characteristics are determined, provide students with graphs of lines that are parallel or perpendicular (several of each). Have students apply the characteristics of parallel or perpendicular lines to write the equations for the given lines. Next, have students write the equation of lines that are parallel or perpendicular to a line through a given point on the line. Examples: Given that a line passes through ( 2,3) and ( ) is parallel to the given line and passes through (, 2). Write the equation of the line that is perpendicular to the original line through (, 2). Solution: Parallel: 5 y = x ; Perpendicular: y = 2x. 2 2 4,6, write the equation of a line that To end the activity, have students working alone to write equations of any two lines that are parallel and any two lines that are perpendicular. Do not provide them with any information such as slopes or y-intercepts. Students should then share the equations with a different partner. Students should graph the equations from their partner to verify that the equations written represent parallel and perpendicular lines. Have students include what they have learned about slopes of parallel and perpendicular lines on the vocabulary cards (view literacy strategy descriptions) they created in Unit 2 Activity to further precisely define those terms. The information concerning the slopes can be placed in the upper left corner with the initial definition of parallel and perpendicular lines. Geometry Unit 3 Parallel and Perpendicular Relationships 3-4
Definition/explanation: Two lines in the same plane that do not intersect; two lines that have the same slope. parallel lines Ways to name: Line AB is parallel to Line CD; Line AB Line CD Real-life objects: Railroad tracks, edges of cereal box, power lines Drawing: A C B D Have students review the cards as they are working with parallel and perpendicular lines. Students can use the cards as a study aid until they can recall the material without the aid of the vocabulary cards. Activity 2: Proving Lines are Parallel (GLEs: 9; CCSS: WHST.9-0.b, WHST.9-0.0) Materials List: pencil, paper, diagrams for discussion, learning log Teacher note: The names of the special angle pairs formed by two lines and a transversal, and the relationships of these special angle pairs are found in the Grade 8 GLEs so they are not discussed in detail here. A review may be necessary depending on the students in the class. Give students diagrams of parallel lines and transversals and diagrams of lines that are not parallel with transversals. Lead a discussion to determine what characteristics of parallel lines will guarantee that two lines are parallel (e.g., two lines are parallel if corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, consecutive interior angles are supplementary, parallel lines are everywhere equidistant). Have students form conjectures that lead to the converses of the parallel lines theorems (e.g., if alternate interior angles are congruent when a transversal intersects two lines, then the two lines are parallel). Remind students that the statement, If two parallel lines are cut by a transversal, then corresponding angles are congruent, is a postulate which is accepted as true without proof. The converse, If two lines in a plane are cut by a Geometry Unit 3 Parallel and Perpendicular Relationships 3-5
transversal so that corresponding angles are congruent, then the two lines are parallel is also a postulate accepted as true without proof. Using these postulates as truth, students can prove the other theorems and converses. Allow students to initially use angle measures to write proofs for specific sets of lines to prove these theorems, but also require them to use general proofs that prove lines parallel through generalities. These proofs can take any form (informal, paragraph, two-column, flow). Provide opportunities for students to prove the other theorems which are based on the postulate for corresponding angles. Have students complete a proof (or proofs) which involves the theorems and/or their converses in their math learning logs (view literacy strategy descriptions). Students should be allowed to complete any type of proof they wish as long as they demonstrate logical thinking and reasoning and provide evidence for their statements. The diagrams should be more than just a pair of parallel lines cut by a transversal. The diagrams used in this proof should incorporate parallel lines in other figures such as triangles and quadrilaterals (in a trapezoid, the angles along one leg are supplementary; opposite angles of a parallelogram are congruent; if a segment is drawn parallel to any side of a triangle, the corresponding angles in the similar triangles are congruent; etc.). Students should not be asked to prove two triangles congruent at this time, but this can serve as a precursor to the proofs they will write in Unit 4. Example Proof: Given: G F H B Prove: D E K Possible Solution: Given the measure of angle FEK, the measure of angle DEF to be 05 degrees can be found because angles DEF and FEK form a linear pair which means the angles are supplementary (80-75 = 05). Using the given measure of angle DFE and the measure of angle DEF, the measure of angle EDF is found to be 45 degrees because angles DFE, EDF, and DEF are the three angles of triangle DFE and the sum of the interior angles of any triangle is 80. Since the measures of angles GFD (given) and EDF are equal, angle GFD is congruent to angle EDF. Angle GFD and angle EDF are alternate interior angles. Since the alternate interior angles are congruent, line GH is parallel to ray DK. Geometry Unit 3 Parallel and Perpendicular Relationships 3-6
Provide diagrams of two lines that are perpendicular to one line. Have students form a conjecture that if two lines are perpendicular to the same line, they must be parallel. Then, have students write a proof of this theorem. To end the activity, have students respond to a SPAWN writing (view literacy strategy descriptions) prompt using the What If? category. Have students answer the following prompt: Think about your favorite outing (a trip to the mall, football games, a trip to Walt Disney World, etc.). What might your day be like if there were no way to ensure lines are parallel? Students responses can be included in their learning logs. After students have had time to construct their responses, allow a few students to share their responses with the class. Students may add their own ideas to other students responses or question their reasoning. These writings could be included in a portfolio of the students work. Activity 3: Distance in the Plane (GLEs: 2, 6; CCSS: RST.9-0.3, WHST.9-0.2d) Materials List: pencil, paper, Distance in the Plane Process Guide BLM, calculator Have students explore the distance between two points in the rectangular coordinate system using the Distance in the Plane Process Guide BLM. This BLM will use a process guide (view literacy strategy descriptions) to guide students through deriving the distance formula from the Pythagorean Theorem. Process guides scaffold students comprehension and are designed to stimulate students thinking during or after their learning. Guides also help students focus on important information and ideas making their learning more efficient. Process guides prompt thinking ranging from simple recall to connecting information and ideas to prior experience, applying new knowledge, and problem-solving. This process guide gives attention to the idea that distance in the plane between two points can be thought of as the length of a hypotenuse of a right triangle. Thus, the Pythagorean theorem can be used to determine these distances. The guide begins by having students recall the Pythagorean Theorem and find missing sides of right triangles. Section B asks students to apply the concept of distance on a number line to find the length of the legs of the right triangle and find the hypotenuse using Pythagorean Theorem. Questions are included to prompt students thinking to assist students in developing the same information. Section C asks students to generalize their work from Section B using arbitrary points ( x, y) and ( x2, y 2) this is where the distance formula is developed. Section D has students apply the distance formula to segments in the coordinate plane. Section E gives students a real-world application to help students connect their learning with a scenario in which they are familiar. When using the Pythagorean Theorem and the distance formula, students are asked to simplify radical solutions as well as use a calculator to estimate the solution. Depending on the level of the students, discussions may need to occur as each section of the process guide is completed to help with any misconceptions. Have students discuss their answers with a Geometry Unit 3 Parallel and Perpendicular Relationships 3-7
partner or a group to check whether they have the correct answers. Once students have completed the process guide, lead a class discussion and have students/groups report their thinking to the class. Have additional examples to provide students with more opportunities to find the distance between points on the coordinate plane. Allow students to use their process guide as needed to work through the additional examples. Encourage students to eventually move from relying on the process guide for guidance to recalling the information from their knowledge. At the end of the lesson, have students reflect on their learning by answering the following prompt in their learning logs (view literacy strategy descriptions): How are the Pythagorean Theorem and the distance formula related? Why would a student choose to use the distance formula over the Pythagorean Theorem? The logs should be collected at some point and read to determine if there are any areas which may need to be clarified. Activity 4: Dividing a Segment Into a Given Ratio on the Number Line (CCSS: G.GPE.6) Materials List: pencil, paper, Dividing Number Line Segments BLM Begin with a discussion (view literacy strategy descriptions) on the following problem: Point D lies on between points A and B and divides into a ratio of 2:3. What is the coordinate of point D? The point of discussion is to help students improve learning and remembering by participating in the dialog about class topics. Class discussion can be used to promote deeper processing of content and rehearsal of newly learned content. For this activity, use the Think Pair Square Share form of discussion. After posing the problem above, give students some time (approximately minute) to think about what the problem means, what they know, what they need to know, and possible ways they may attempt to answer the problem. After that time has passed, have students pair up and share their ideas with each other. Student A should take minute to share his/her ideas and then student B should be given minute to share his/her ideas. Time should also be given for the students to talk briefly and ask each other questions to clarify their thoughts if necessary. The third step is to have the pairs of students share with another pair (four students, or a square). Again, each pair should be given time to give their ideas to the other pair and vice versa. After the square has been given time to discuss, have the students report to the entire class. Record students ideas for reference throughout the lesson. After the discussion, provide students with a copy of the Dividing Number Line Segments BLM. Students should be guided through completing the Dividing Number Line Segments BLM. Using the responses from the discussion, begin to identify the correct understandings and any misunderstandings about the problem. Guide students to understand that in order to find a point that divides the segment into a ratio of 2:3, there must be five equal parts. Point D would then be 2 of the five parts (or 2/5 of the distance) from point A and 3 of the five parts (or 3/5 of the distance) from point B. Using the number line drawn on the board, demonstrate how can be physically divided into 5 Geometry Unit 3 Parallel and Perpendicular Relationships 3-8
equal parts, and then find the correct location of D. Next, show students how they can find the point analytically (this is question 4 on the Dividing Number Line Segments BLM) first, find 2/5 of the length of, then add that measure to the coordinate for A. ; D is 4 units from A so the coordinate of D is - + 4 = 3. Be sure to point out that since the students are finding 2/5 of 0, they are finding the distance D is from A as opposed to the coordinate of D since 0 is the length of the segment. Provide students with a few more examples as needed, then have students work problems 5 and 6 on the Dividing Number Line Segments BLM for independent practice. After the answers to 5 and 6 have been discussed, complete number 7 as a whole class. Elicit student responses to help develop the parts of the formula and be sure they understand what each variable represents. Note: The absolute value in the equation forces the distance to be positive, and is important in this formula. In the next activity, the absolute value will be removed because the direction of the line will produce a signed distance (which means the expression for the distance may result in a negative value) that will be necessary to divide the segments on the coordinate plane correctly. Give students a few more examples and have them use the formula they just developed to find the desired point. Be sure to revisit all ideas created through the discussion at the beginning of the lesson and clarify any misconceptions or incorrect approaches to this skill. As a closure, have students explain what the formula means, including what each variable represents. This explanation can be written in their learning logs (view literacy strategy descriptions) which can be collected and/or shared with the class. Activity 5: Dividing a Directed Line Segment Into a Given Ratio on the Coordinate Plane (CCSS: G.GPE.6) Materials List: pencil, paper, graph paper (optional) This activity will extend the concept of dividing a segment into a given ratio on the number line to the coordinate plane. If the given segment is horizontal or vertical, students can use the same intuitive thinking developed in Activity 4. However, for directed line segments, or segments with a non-zero slope, the formula for finding the point is not as intuitive when working with the coordinate plane, so the formula developed in Activity 4 will be extended. The formula developed used the distance between the endpoints on the number line. For directed segments, the formula will use the signed distance this means students will not take the absolute value of the distance to find the ordered pair. Review with students how to divide a segment into a given ratio on a number line. k Remind students of the formula developed in Activity 4 ( C A) + A. Pose the k+ k2 following problem: Given AC, where A( 3, ) and C ( 2,5), find B such that B divides AC into a ratio of :2. Have students think about their responses for about 30 Geometry Unit 3 Parallel and Perpendicular Relationships 3-9
seconds, then share with a partner. Next, have students report to the class their thoughts and use their responses to attempt to solve the problem. Guide them to use the formula for the x-coordinates and the y-coordinates separately. Solution: X B = 2 3 + 3 3 = ( 5 ) + ( 3 ) 3 5 = 3 3 4 = or 3 3 ( ( ) ) ( ) Y B = ( 5 ( ) ) + ( ) 3 = ( 6 ) + ( ) B: 3 = 2 =, 3 Have students graph the points to see if their answer is reasonable. Then have students find the lengths of segment AB and segment BC and verify that the ratio of AB:BC is :2. Solution: 6 2 6 AB = and BC = 3 3 Next, pose the following problem to the students: Given RT, where R( 3, 4) and T ( 2, 6), find S such that S divides RT into a ratio of 2:3. Remind students that A in the formula should always be represented by the leftmost point. Have students use the formula developed in Activity 4 to attempt to find S. Students should graph the point they find to determine if it is reasonable. They will find that the point they will obtain is NOT on the line. Ask students for their thoughts as to why that happened. Some students may realize that they could subtract from the y-coordinate of R and get the answer. Others may realize that they could find 3/5 of 0 (the distance from Y R to Y T ) and add it to the y- coordinate of T. Help students understand that this formula is not efficient to find this point. Have students work the problem again, but this time do not use the absolute value. S, 0. Ask students to identify what is different about this line Students should get ( ) segment from AC. Students should respond that segment RT has a negative slope while segment AC has a positive slope. Explain to students that the direction is important to help determine the correct placement of S and that removing the absolute value uses the signed distance to determine the coordinates of S. Redefine the formula as stated below: Given any AB and point P such that P divides AB into the ratio k :k 2, the coordinates of P are k xp = ( xb xa) + xa and yp = k ( yb ya) + ya, where ( xa, y a) is the k+ k2 k+ k2 leftmost point on the graph. Give students more examples and have them verify that the ratio of the lengths of the two smaller segments equal the given ratio. Geometry Unit 3 Parallel and Perpendicular Relationships 3-0
Activity 6: Parallel Lines and Distance (GLEs: 6; CCSS: WHST.9-0.2d, WHST.9-0.0) Materials List: pencil, paper, computer drawing program (optional), learning log Provide different sets of two lines. Some sets should be parallel; others should not be parallel. These lines should be drawn on lineless paper. Ask students how they visually determine which of the sets of two lines are parallel. Have a discussion which leads to an understanding that parallel lines are always the same distance apart. Lead students in a discussion about the definition of distance between a point and a line or distance between two parallel lines. One way to do this is to draw two parallel lines on the board and use a ruler to determine distances. Use a drawing program, such as Geometer s Sketchpad, as an alternate way to demonstrate the same concept. The discussion should reveal that the distance is always the shortest line segment between two points (or the shortest distance between a point and a line). Have students realize that the distance between a point and a line is the same as the length of a line segment which starts at the point and is perpendicular to the line. To find the distance between two parallel lines, identify a point on one of the lines and draw a segment from this point perpendicular to the second line. Give students diagrams on the coordinate plane and ask them to find the distance between lines and points not on the lines. Review the concept of the Parallel Postulate here (If there is a line and a point not on the line, then there is exactly one line that can be drawn through the given point that is parallel to the given line). Have students apply the concept of distance between a point and a line to polygons. Relate the distance between a point and a line to finding the length of an altitude in a triangle (i.e., an altitude is the perpendicular distance from a vertex to a segment on the opposite side of the triangle). This establishes correct understandings necessary for the concepts that will find area, surface area, and volume. To end the activity, have the students answer the following prompt in their math learning logs (view literacy strategy descriptions): Describe how to determine the distance between a line and a point not on the line. How could you use this information to help you find the distance between a plane and a point that is not contained in that plane? Geometry Unit 3 Parallel and Perpendicular Relationships 3-
Activity 7: Parallel Line Facts (GLEs: 9; CCSS: WHST.9-0.b) Materials List: pencil, paper, Parallel Line Facts BLM, ruler Teacher Note: The focus of this activity is to apply the properties of parallel lines learned in earlier activities. This is a good application of proof using the parallel line properties. It also introduces information relative to angle relationships in triangles, connecting this unit to Unit 4. Give students copies of the Parallel Line Facts BLM and have them complete the following steps. Have students draw a line through one vertex of a triangle so that the line is parallel to a side of the triangle. Have students write a proof (using parallel line relationships from Unit 2 and earlier activities) to show that the sum of the angles in a triangle is 80. Use the same diagram to write a proof to show that the measure of an exterior angle in a triangle has the same measure as the sum of the two remote interior angles. This the Exterior Angle Sum theorem. Have students investigate the area of different triangles formed between two parallel lines by moving one vertex along one of the parallel lines. Students should recognize that the height of each triangle is always the same since the distance between the parallel lines will not change. Since the base length doesn t change, students should realize that the areas are the same. Have students develop a proof of this as well. Sample Assessments General Assessments The student will create a portfolio containing samples of work completed during activities. For instance, he/she could include the graphs from Activity 2 and explain what happened in the activity and what was learned from the activity. The student will respond to journal prompts that include: o Describing at least three different ways to prove two lines are parallel. 2 o Explaining how to write the equation of a line perpendicular to y = x+ 5 3 through the given point (-4,6). o Explaining the relationship between the Pythagorean theorem and the distance formula for distance on the coordinate plane. The student will create a scrapbook of pictures taken in a real-world setting (i.e. railroad tracks) that depict parallel and perpendicular lines. This scrapbook will include pictures and indicate how the items in the picture demonstrate the term Geometry Unit 3 Parallel and Perpendicular Relationships 3-2
chosen. The student will have a minimum of three pictures for each term. See the Scrapbook Rubric BLM for more information. Activity-Specific Assessments Activity : The teacher will provide the student with several sets of graphs of perpendicular lines, intersecting lines which are not perpendicular, and intersecting lines which are almost perpendicular drawn on coordinate graph paper. The student will use slope to determine if the lines are parallel. Activity : The teacher will give each student a copy of the What s My Line? BLM and one of the five What s My Line? graphs. The five graphs have different slopes, so there is a larger probability that the students will have different responses. The student will draw and label the x- and y-axes anywhere on the coordinate plane that he/she chooses. Based on where the x- and y-axes are drawn, the student will then o find the slope of the given line and write the equation of the given line. o write the equation for a line which passes through the given point and is parallel o to the given line. o write the equation of the line which is perpendicular to the given line and passes o through the given point. A What s My Line? Rubric BLM is provided for this assessment. The What s My Line? BLM, the What s My Line? Graphs, and the What s My Line? Rubric BLM are located at the end of the BLMs for Unit 3. Activities, 3, and 5: The student will solve constructed response items such as this: Point S is located at (-5,7) and point T is located at (-,-). a. Graph the segment. b. Find the length of segment ST. ( ) c. Find the coordinates of R such that R divides segment ST into a ratio of :3. (R(-4,5)) d. Verify that the ratio of SR:RT is :3 using the distance formula. e. Find the equation of the line perpendicular to segment ST that passes through R and graph the line. ( y = x+ 7 ) 2 Geometry Unit 3 Parallel and Perpendicular Relationships 3-3