6.00 Trigonometry Geometry/Circles Basics for the ACT. Name Period Date

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1 6.00 Trigonometry Geometry/Circles Basics for the ACT Name Period Date Perimeter and Area of Triangles and Rectangles The perimeter is the continuous line forming the boundary of a closed geometric figure. The area is the number of square units that covers a closed geometric figure. Ex: Find the perimeter and the area of the geometric figures. Square: Perimeter: Because all the sides are the same, the perimeter will be 4 times the length of one side. P 4(6.3) 5.km Area: The area will a side times a side, or a side. A (6.3) 39.69km Rectangle and Parallelogram: P (3) (0) 6cm Perimeter: Add the lengths of all four sides. A cm Because opposite sides of a rectangle and a parallelogram are equal, you can use the formula: P = l +w, where l is the length and w is the width. Area: In a rectangle, to find area you just multiply the length and the width: A = lw. It is a little trickier with a parallelogram, because to find area, the length and width must be perpendicular to one another (form a right P (3.7) angle.) The sides of a parallelogram are not necessarily perpendicular. Therefore you must take the base x height, A 7. where the height is the perpendicular distance between the bases. (7.).6yd 4.yd Triangle: Perimeter: Add the three sides. No special formula. Area: Once again you must use the perpendicular base and height. But you also must remember P 7 9 8km to multiply by ½! A 5. 3.km

2 Trapezoid: Perimeter: Add the four sides. No special formula. Area: A trapezoid has TWO bases the parallel sides. Therefore, you must add the bases together, then multiply by ½, and then multiply by the height, which is the perpendicular distance between the two parallel bases. P cm A (3.5.9) cm To find the missing value when you know the area, use the formula, plug in the values you know, and follow the rules of algebra to solve for the missing value. Ex: Find the missing value with the given information. A bh 5.8 ( b)(6) 5.8 3b 8.6 b A 64.8 ( b ( b 3.4) b 3.4 b ( b b ) h 3.4) 9 Compute Circumference and Area of Circles Perimeter: The perimeter of a circle is called the circumference. Use the formula: C = π diameter, or if you know the radius, use C = π radius Area: Use the formula: A = π radius. **On the ACT, you will usually leave your answers in terms of π because the test was actually designed to be completed without a calculator! Ex: Find the circumference and area of the circle: C r C () A r A () C ft A ft

3 Perimeter and Area of Simple Composite Geometric Figures The perimeter of an irregular shape is equal to the sum of the segments and semi-circles around the outside of the figure. A composite figure is made up of several simple geometric figures such as triangles, rectangles, squares, circles, and semicircles. To find the area of a composite figure, separate the figure into simpler shapes whose area can be found. Then add the areas together. Example: Find the perimeter and area of the composite figure: Perimeter: the outside is made up of two 60 yard segments, and two semi-circles who diameters are 60. If put together, they would form a circle with radius 30 yds. Find the circumference P of the circle and add it to the two 60 yd. sides. P 0 60 yds Area: Find the area of the 60 yd. square, and the area of the circle with a diameter of 60. Add them together. A s A 60 r (30) A yds Arc Length & Sector Area in Circles The arc length is the measure of the distance along the curved line making up the arc. It is longer than the straight line distance between its endpoints (which would be a chord). A minor arc is named by its two endpoints, such as AB. An arc > 80 o is a major arc and is named by its endpoints and one point inbetween, such as ADB. An arc = 80 o is a semi-circle. A circular sector or circle sector (symbol: ), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. Ex: Find the exact arc length. 05 Solution: the length of the arc is of the circumference of the 360 circle. Use this information to write an equation and solve. Arc length mi

4 Ex: Find the exact sector area. 5 Solution: the area of the sector is of the area of the entire 360 circle. Use this information to write an equation and solve. Angles in Circles Sector Area km There are many relationships between the angles, chords, secants, and tangents of a circle. The most commonly used on the ACT are: Central Angles central angles have their vertex at the center of the circle. A central angle s measure is equal to the arc it subtends (cuts through.) 88 o Arc FD = 8 o Inscribed Angles inscribed angles have their vertex ON the circle. An inscribed angle is ½ of the arc it subtends. Tangents any tangent to a circle meets the radius at the point of tangency at a 90 o angle. Because of the right angle, right triangles can be formed, which means the Pythagorean Theorem can be used to find missing measurements. Vertical angles formed in circles are still congruent; Linear Pairs are still supplementary. Pythagorean Theorem/Pythagorean Triples For any right triangle, the length of the hypotenuse squared (the longest side, opposite the right angle) is equal to the sum of the squares of the two legs of the right triangle. Pythagorean Theorem: a + b = c The converse of the Pythagorean Theorem is also true: If three sides of a triangle work in the Pythagorean Theorem, then they are the sides of a right triangle. If the side lengths are integers, it is said they form a Pythagorean Triple.

5 Examples: Use the properties of the angles of a circle and the Pythagorean Theorem to find the exact (in terms of π) answers to the following: Solution: The arcs are the same as the central angles. Because the central angles are vertical, they are congruent. Therefore the arcs are also congruent. Set up an equation and solve. x x 6 x 6 x 6 The angle is an inscribed angle, therefore it is ½ of the arc LM. Set up an equation and solve. 7x 4 (3x ) 7x 4 6x x 5 m LKM 7(5) 4 m LKM 8 o The line in question is tangent to the circle, which means it forms a right angle with the radius at that point. Since we have a right triangle and know sides, we can use the Pythagorean Theorem to find the missing side. (Careful the one side is a diameter of the circle, so it is 6, not 8!!) a a a a a b a c 0 44

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h r c On the ACT, remember that diagrams are usually drawn to scale, so you can always eyeball to determine measurements if you get stuck.

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