B. Dilate the following figure using a scale

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1 1 Dilations affect the size of the pre-image. he pre-image will enlarge or reduce by the ratio given by the scale factor. A dilation with a scale factor of k > 1 enlarges it. A dilation of 0 < k < 1 reduces it. 1 You ry: A. Dilate the following figure using a scale factor of 2 with center of dilation at the origin. Ex: Dilate the following figure using a scale factor of 3 with center of dilation at (5,-6). B. Dilate the following figure using a scale One Solution: Plot (5, 6). Draw lines from the center of dilation through vertices of the pre-image. Since the scale factor is 3, each distance from the center of dilation to the image will triple. Plot image s vertices and connect them to complete the image. factor of 1 2 with center at (4,-2). C. Dilate the following figure using a scale factor of 3 with center of dilation at the origin. or all dilations centered at (a, b) with a scale factor of k, the image s coordinates can be found using ( a + k( x a), b + k( y b)). If the center of dilation is at the origin, then a and b are zero, resulting in the new image location coordinates as ( k a, k b). G.SR.1 Page 1 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

2 2 When a figure is dilated to make an image, corresponding angles are equal and corresponding sides are proportional relative to the scale factor used to dilate. wo different-sized figures can be shown to be similar by using transformations if one of the figures can be mapped onto the other using a series of transformations, one of which is a dilation and the other(s) a reflection, rotation and/or translation. Ex #1: Prove the following figures are similar by describing a series of transformations that will map igure 1 onto igure 2. 2 A. Prove the following figures are similar by describing a series of transformations that will map the smaller triangle to the larger triangle. B. Are these triangles similar? Justify your reasoning. One possible solution: Dilate igure 1 by a scale factor of 2 with center of dilation at ( 3, 2). hen translate the resulting image four units right and four units down. Ex #2: If ~, find CD. Solution: Since the figures are similar, then a dilation has occurred using a scale factor. his creates corresponding sides that are proportional: BD G CD HG 30 6 CD G.SR.2 Page 2 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

3 3 Phillip draws two triangles. wo pairs of corresponding angles are congruent. Select each statement that is true for all such pairs of triangles. A. A sequence of rigid motions carries one triangle onto the other. 3 Omar thinks that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. o show this, he drew the figure below. B. A sequence of rigid motions and dilations carries one triangle onto the other. C. he two triangles are similar because the triangles satisfy Angle-Angle criterion. D. he two triangles are congruent because the triangles satisfy Angle-Angle criterion. E. All pairs of corresponding angles are congruent because triangles must have an angle sum of All pairs of corresponding sides are congruent because of the proportionality of corresponding side lengths. Which set of transformations maps ABC to DEC and supports Omar s thinking? Solutions: rue B, C, and E. A is only true for congruent triangles and not for similar triangles. D is not true because Angle-Angle does not prove triangle congruency. is not true because corresponding sides are not congruent for similar triangles. A. A rotation of 180 clockwise about point C followed by a dilation with a center of point C and a scale factor of 2. B. A rotation of 180 clockwise point C followed by a dilation with a center of point C and a scale factor of 1 2. C. A rotation of 180 clockwise point C followed by a dilation with a center of point C and a scale factor of 3. D. A rotation of 180 clockwise point C followed by a dilation with a center of point C and a scale factor of 1 3. G.SR.3 Page 3 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

4 4 Ex #1: A flagpole 4 meters tall casts a 6-meter shadow. At the same time of day, a nearby building casts a 24-meter shadow. How tall is the building? Solution: Draw a picture: 4 A. Mark stands next to a tree that casts a 15- foot shadow. If Mark is 6 feet tall and casts a 4-foot shadow, how tall is the tree? 4 m h 6 m 24 m Write a proportion and solve: h h 4 h 4 4 ( 4) 4( 4) h 16 he building is 16 meters tall. B. ind the value of x in the figure below. Ex #2: ind BE. Let BE x. Or using the Parallel/Proportionality Conjecture: 4 x 8 9 ( ) ( x) x 4.5 x G.SR.5 Page 4 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

5 5 Similar right triangles have side ratios that are equal to each other. or example, every triangle, no matter what size, has a small side to hypotenuse ratio of 1:2 or 1 (or 0.5). 2 hese are the side length ratio definitions of the acute angle, θ : sinθ Opposite Hypotenuse 5 Given MA, match each trigonometric ratio to its equivalent value in the box. M A 8 cosθ tanθ Adjacent Hypotenuse Opposite Adjacent Ex: Write each trigonometric ratio using the side lengths of ABC below. 1) cos 2) cos M 3) tan M 4) tan 5) sin M 6) sin 8 A. 17 B C D. 15 A. sin C B. cosc C. tan A D. tan C E. cos A Solutions: Answers: A. 4 5 B. 3 5 C. 3 4 D. 4 3 E. 4 5 G.SR.6 Page 5 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

6 6 he sine of an angle is equal to the cosine of its complement: sinθ cos(90 θ). Q 6 he table below shows the approximate values of sine and cosine for selected angles. P According to the figure above: sin P and cosr So, sin P cosr If sin , then the cosine of its complement is equivalent: sin32 cos(90 32 ) cos R A. ill in the rest of the table without a calculator. Angle Value of Sine Value of Cosine B. Explain how you determined the values you used. Ex: Determine whether the following statements are true or false: 1. sin 43 cos47 2. sin 43 cos43 3. sin 45 cos45 4. sin17 cos(90 17) 5. cosθ sin(90 θ) Solutions: 1 rue, 2 alse, 3 rue, 4 rue, 5 rue G.SR.7 Page 6 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

7 7 Drawing and labeling pictures are a great way to solve problems using the trigonometric ratios. Don t forget the Pythagorean heorem ( + )! he angle of elevation from a landscaped rock to the top of a 30-foot tall flagpole is 57. Which of the following equations could be used to find the distance between the rock and the base of the flagpole? Select all that apply. 7 A plane is flying at an elevation of 900 meters. rom a point directly underneath the plane, the plane is 1200 meters away from a runway. Select all equations that can be used to solve for the angle of depression (θ) from the plane to the runway. A. 30 sin 57 x B. cos 57 C. x tan 57 x D. tan 33 Solution: x 30 A. B. C. D. E. 900 sinθ cosθ sinθ cosθ tanθ 1200 All the angles in a triangle add up to 180, so the acute angle near the flag at the top of the triangle must measure 33. A and B are not correct because the ratios do not correspond to the definitions of the trig ratios. C and D are correct since the tangent ratios of those opposite angles do show hypotenuse. G.SR.8 Page 7 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

8 8 Solve for the variables. 8 J a K his is a triangle. Using theorem: leg leg x 9 hypotenuse legi y 9i 2 y Using leg:leg:hyp. ratios: leg leg 1 x 1 9 x 9 1:1: 2 leg hyp y y 9 2 G.SR.8.1 Determine whether each of the following statements is true or false. A. a b b L B. JKL is a right scalene triangle C. Area of JKL 25 2 sq. un. D. Perimeter of JKL ( + ) un. 9 Solve for the variables. m 9 Y n p 60 n his is a triangle. Using theorem: hypotenuse 2 i 24 2m 12 m ( short leg ) ( long leg ) ( short leg ) i 3 n 12i 3 n 12 3 Using short leg:long leg:hypotenuse ratios: short leg hypotenuse 1 m m 24 m 12 1: 3 : 2 long leg hypotenuse 3 n n 24 3 n 12 3 G.SR.8.1 X Determine whether each of the following statements is true or false. A. m X 30 B. n 6 C. n 6 3 D. p E. Area of XYZ 24 3 sq. un. End of Study Guide Z Page 8 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

9 1 You ry Solutions: A. Dilate the following figure using a scale factor of 2 with center of dilation at the origin. 2 A. Prove the following figures are similar by describing a series of transformations that will map the smaller triangle to the larger triangle. OR multiply each vertex s coordinate by the scale factor of 2 to find the image s coordinates: ( 1, 1) (2 1, 2 1) ( 2, 2) ( 2, 3) (2 2, 2 3) ( 4, 6) (0, 3) (2 0, 2 3) (0, 6) One solution could be dilating ABC by a scale factor of 3 with center of dilation at (2, 1) and then translated 4 units right and 2 units up, then ABC maps onto A' B ' C '. B. Dilate the following figure using a scale factor of 1 2 with center at (4,-2). Another solution could be dilating ABC by a scale factor of 3 with center of dilation at the origin. B. Are these triangles similar? Justify your reasoning. C. Dilate the following figure using a scale factor of 3 with center at the origin. If the triangles are similar, then all corresponding side ratios must be equal since a dilation has occurred ~. Page 9 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

10 3 Omar thinks that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. o show this, he drew the figure below. 4 A. Mark stands next to a tree that casts a 15-foot shadow. If Mark is 6 feet tall and casts a 4-foot shadow, how tall is the tree? h 6 ft Which set of transformations maps ABC to DEC and supports Omar s thinking? he scale factor is 2 since the corresponding sides have a ratio of 6:3, or 2:1. herefore, A is the correct answer. 15 ft h h 15 (4)(6) (4)(6) 6 4 4h 90 h 22.5 he tree is 22.5 ft high. 4 ft B. ind the value of x in the figure below. ( x + 2) x x (24) (24) ( x + 10) 34 3x x 4 4 x 3 Page 10 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

11 5 6 Given MA, match each trigonometric ratio to its equivalent value in the box. M 1. A 2. B 3. D 4. C 5. A 6. B 15 he table below shows the approximate values of sine and cosine for selected angles. 17 A. ill in the rest of the table. G.SR.6 Angle Value of Sine Value of Cosine B. Explain how you determined the values you used. he sine of an angle is equal to the cosine of its complement. So, sin15 cos 75, sin 30 cos 60, sin 45 cos 45 and sin 0 cos90. A 8 7 A plane is flying at an elevation of 900 meters. rom a point directly underneath the plane, the plane is 1200 meters away from a runway. Select all equations that can be used to solve for the angle of depression (θ) from the plane to the runway. angle of depression Alternate Interior Angles Solution: Use the Pythagorean heorem to find the length of the hypotenuse: + (1200) +(900) c Based on this information, the following are equations that can be used to solve for the angle of depression: A. B. E. θ 1200 m 900 sinθ cosθ tanθ 1200 θ 900 m Page 11 of 12 MCC@WCCUSD (WCCUSD) 12/17/15

12 8 Determine whether each of the following statements is true or false. A. a b B. JKL is a right scalene triangle C. Area of JKL 25 2 sq. un. D. Perimeter of JKL ( + ) un. 9 Determine whether each of the following statements is true or false. A. m X 30 B. n 6 C. n 6 3 D. p 8 3 E. Area of XYZ 24 3 sq. un. Page 12 of 12 MCC@WCCUSD (WCCUSD) 12/17/15