Slide 1 / 53
Slide 2 / 53 7th Grade Drawing Geometric Figures 2015-11-23 www.njctl.org
Slide 3 / 53 Topics Table of Contents Determining if a Triangle is Possible Click on a topic to go to that section Geometric Constructions: The Basics Glossary Videos Table of Contents Videos Using Geometer's Sketchpad Video: Constructing Circles Video: Constructing Isosceles Triangles Video: Constructing Equilateral Triangles Video: Congruent Triangles
Slide 4 / 53 Determining if a Triangle is Possible Return to Table of Contents
Slide 5 / 53 Drawing Triangles How many different acute triangles can you draw? How many different right scalene triangles can you draw? Recall that triangles can be classified according to their side lengths and the measure of their angles. Sides: Scalene - no sides are congruent Isosceles - two sides are congruent Equilateral - all three sides are congruent Angles: Acute - all three angles are acute Right - contains one right angle Obtuse - contains one obtuse angle
Slide 6 / 53 Triangle Inequality Property There is another property that applies to triangles: Click the lab below to determine what the Triangle Inequality Property is about. Triangle Inequality Lab
Slide 7 / 53 Triangle Inequality Property Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What does this mean? If you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. If that is not true, then it is not possible to construct a triangle with the given slide lengths.
Slide 8 / 53 Example Determine if sides of length 5 cm, 8 cm and 12 cm can form a triangle? Test all three pairs to see if the sum is greater: 5 + 8 > 12 5 + 12 > 8 8 + 12 > 5 13 > 12 17 > 8 20 > 5 Yes, it is possible to construct a triangle with sides of lengths 5 cm, 8 cm and 12 cm.
Slide 9 / 53 Example Determine if sides of length 3 ft, 4 ft and 9 ft can form a triangle? Test all three pairs to see if the sum is greater: 3 + 4 > 9 3 + 9 > 4 4 + 9 > 3 7 > 9 12 > 4 13 > 3 No, it is not possible to construct a triangle with sides of lengths 3 ft, 4 ft and 9 ft.
Slide 10 / 53 Try These Determine if triangles can be formed with the following side lengths: 1. 4 cm, 7 cm, 10 cm 2. 24 mm, 20 mm, 30 mm 4 + 7 > 10 24 + 20 > 30 4 + 10 > 7 24 + 30 > 20 7 + 10 > 4 20 + 30 > 24 YES YES 3. 7 ft, 9 ft, 16 ft 4. 9 in, 13 in, 24 in 7 + 9 = 16 9 + 13 < 24 7 + 16 > 9 9 + 24 > 13 16 + 9 > 7 13 + 24 > 9 NO NO
Slide 11 / 53 1 Determine if sides of length 5 mm, 14 mm and 19 mm can form a triangle. Be prepared to show your work! Yes No
Slide 12 / 53 2 Determine if sides of length 6 in, 9 in and 14 in can form a triangle. Be prepared to show your work! Yes No
Slide 13 / 53 3 Determine if sides of length 5 yd, 13 yd and 21 yd can form a triangle. Be prepared to show your work! Yes No
Slide 14 / 53 4 Determine if sides of length 3 ft, 8 ft and 15 ft can form a triangle. Be prepared to show your work! Yes No
Slide 15 / 53 5 Determine if sides of length 5 in, 5 in and 9 in can form a triangle. Be prepared to show your work! Yes No
Slide 16 / 53 6 A triangle could have which of the following sets of angles? A 40º, 90º, 105º B 35º, 89º, 56º C 75º, 90º, 15º D 30º, 65º, 95º
Slide 17 / 53 7 A triangle could have which of the following sets of angles? A 37º, 63º, 80º B 90º, 104º, 76º C 23º, 47º, 50º D 80º, 90º, 10º
Slide 18 / 53 Example Predict the length of the third side of a triangle with sides of 12 ft and 16 ft. length Side 1 = 12 ft Side 2 = 16 ft The 3rd side must be less than: 12 + 16 > 3rd side 28 ft > 3rd side The 3rd side must be greater than: 12 + 3rd side > 16 3rd side > 4 The 3rd side must be greater than 4 ft and less than 28 ft.
Slide 19 / 53 Example Predict the length of the third side of a triangle with sides of 9 cm and 15 cm. length Side 1 = 9 cm Side 2 = 15 cm The 3rd side must be less than: 9 + 15 > 3rd side 24 cm > 3rd side The 3rd side must be greater than: 9 + 3rd side > 15 3rd side > 6 The 3rd side must be greater than 6 cm and less than 24 cm.
Slide 20 / 53 Try These Predict the length of the third side of a triangle whose known sides are lengths: 1. 13 mm, 20 mm 2. 7 in, 19 in 13 + 20 > Side 3 7 + 19 > Side 3 33 > Side 3 26 > Side 3 13 + Side 3 > 20 7 + Side 3 > 19 Side 3 > 7 Side 3 > 12 7 < side 3 < 33 12 < side 3 < 26
Slide 21 / 53 Try These Predict the length of the third side of a triangle whose known sides are lengths: 3. 4 ft, 11 ft 4. 23 cm, 34 cm 4 + 11 > Side 3 23 + 34 > Side 3 15 > Side 3 57 > Side 3 4 + Side 3 > 11 23 + Side 3 > 34 Side 3 > 7 Side 3 > 11 7 < side 3 < 15 11 < side 3 < 57
Slide 22 / 53 8 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
Slide 23 / 53 9 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
Slide 24 / 53 10 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
Slide 25 / 53 11 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
Slide 26 / 53 12 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
Slide 27 / 53 13 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
Slide 28 / 53 Geometric Constructions: The Basics Return to Table of Contents
Slide 29 / 53 Geometric Tools In Geometry, we can draw just about every figure with various tools. The tools that we will be using are given on this slide & the next slide: 1) Compass : creating circles & arcs
Slide 30 / 53 2) Ruler : measure segments Geometric Tools D E DE = 6 cm 3) Protractor: measure angles A m ABC = 65 B C
Slide 31 / 53 Example Draw a circle that has a radius of 6 cm. Step #1: Draw a segment w/ the ruler that measures 6 cm. Step #2: Line up your compass so that it's center tip lies on one endpoint & the pencil tip lies on the other endpoint.
Slide 32 / 53 Step #3: Keeping the distance between the center & endpoint the same, draw your circle.
Slide 33 / 53 Try this! 1) Construct a Circle that has a radius of 3 cm using a ruler & a compass.
Slide 34 / 53 Try this! 2) Construct a Circle that has a radius of 8 cm using a ruler & a compass. Video: Constructing Circles
Slide 35 / 53 Example Use a ruler & a compass to draw an isosceles triangle w/ the following conditions: 1. at least one of the sides is 7 cm 2. at least one of the sides is 3 cm Step #1: Look at your conditions. Both of them say "at least one" which means that one side, or more sides could meet the conditions. Plus, since the triangle is isosceles, we know that two of the sides must be equal. So pick which number you want to occur for 2 of your sides. I'll select the 7 cm to occur twice. Step #2: Draw one of your 7 cm segments. A B
Slide 36 / 53 Step #3: With your compass, draw a circle or semicircle (whichever you prefer). A B Any segment that I connect from this arc to the center will have the same radius length of 7 cm.
Slide 37 / 53 Step #4: With your ruler, find the segment that can be drawn from B to another point on the semicircle so that the ruler measures 3 cm. Make a point at this location. A B Step #5: Connect this point with points A & B to form your triangle. C A B Note: AC = AB = 7 cm, since they are both radii of the circle.
Slide 38 / 53 Think about this... Can we make a triangle if we use the 3 cm twice and the 7 cm once? Discuss this problem in your groups for a few minutes.
Slide 39 / 53 Try this! 3) Use a ruler & a compass to draw an isosceles triangle w/ the following conditions: 1. at least one of the sides is 8 cm 2. at least one of the sides is 6 cm Video: Constructing Isosceles Triangles
Slide 40 / 53 Example Use a ruler & a protractor to draw an equilateral triangle that has a side length of 5.5 cm. Step #1: Reread your question. We have an equilateral triangle, which has all of the sides measuring 5.5 cm & all of its angles measure to be 60. Step #2: Draw one of your 5.5 cm segments. C D
Slide 41 / 53 Step #2: With your protractor, place its center at either point C or point D (doesn't matter), measure 60 & draw a line connecting the 60 measurement with your center (mine is C). C D Step #3: Repeat step #2 with your other endpoint as the center. C D
Slide 42 / 53 Step #4: Draw a point where the 2 lines intersect & erase any additional lines. C D Note: You can verify that all of the edges are equal by measuring all of them.
Slide 43 / 53 Try this! 4) Construct an equilateral triangle that has a side length of 4.2 cm using a protractor and a ruler.
Slide 44 / 53 Try this! 5) Construct an equilateral triangle that has a side length of 6.7 cm using a protractor and a ruler. Video: Constructing Equilateral Triangles
Slide 45 / 53 Example Use a ruler & a protractor to draw FGH that satisfies the following conditions. 1. m F = 30 2. m G = 70 3. FG = 8 cm Step #1: Reread your question. We know two angle measurements and the length of 1 side. Let's start with the side, since at least 1 segment is required to use a protractor. Step #2: Draw your 8 cm segment. Label one endpoint with F & the other with G F G
Slide 46 / 53 Step #2: With your protractor, place its center at point F. Measure the 30 required for m F. Draw a line through the 30 angle F G Step #3: With your protractor, place its center at point G. Measure the 70 required for m G. Draw a line through the 70 angle F G
Slide 47 / 53 Step #4: Draw point H where the 2 lines intersect & erase any additional lines. H F 30 70 8 cm G
Slide 48 / 53 Try this! 6) Construct JKL with the given conditions using a protractor and a ruler. 1. m K = 105 2. m L = 25 3. KL = 9 cm Video: Constructing Triangles
Slide 49 / 53 Glossary Return to Table of Contents
Slide 50 / 53 Compass An instrument with 2 arms, one sharp & one with a pencil that can be used to draw circles and arcs. Back to Instruction
Slide 51 / 53 Protractor An instrument used to measure angles. They are usually half of a circle (180 ). To measure an angle, line up the protractor so that the vertex of the angle aligns with the small hole, or dot, near the bottom of the protractor and one of the angle edges lies on the straight line at the bottom. 30 135 90 Back to Instruction
Slide 52 / 53 Ruler An instrument used to measure the lengths of segments. E AB = 3.5 cm A B C CD = 5 cm EF = 6.5 cm D F Back to Instruction
Slide 53 / 53 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 5 + 3 > 10 6 + 4 > 10 7 + 5 > 10 5 10 3 6 10 4 7 10 5 5 10 3 6 10 4 7 10 5 Back to Instruction