You Try: A. Dilate the following figure using a scale factor of 2 with center of dilation at the origin.

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Jerome Webster
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1 1 G.SRT.1-Some Tings To Know Dilations affect te size of te pre-image. Te pre-image will enlarge or reduce by te ratio given by te scale factor. A dilation wit a scale factor of 1> x >1enlarges it. A dilation of 1< x <1reduces it. 1 You Try: A. Dilate te following figure using a scale factor of 2 wit center of dilation at te origin. Ex: Dilate te following figure using a scale factor of 3 wit center of dilation at (,-6). B. Dilate te following figure using a scale factor of 1 2 wit center at (4,-2). One Solution: Plot (, 6). Draw lines from center of dilation troug vertices of te preimage. Since te scale factor is 3, eac distance from te center of dilation to te image will triple. Plot image s vertices and connect tem to complete te image. C. Dilate te following figure using a scale factor of -2 wit center of dilation at te origin. For all dilations centered at (a, b) wit a scale factor of k, te image s coordinates can be found using (a + k(x a), b + k(y b)). If te center of dilation is at te origin, ten a and b are zero, resulting in te new image location coordinates as (k a, k b). G.SRT.1 G.SRT.1 Page 1 of 11 MCC@WCCUSD 12/11/14

2 2 G.SRT.2-Some Tings To Know Wen a figure is dilated to make an image, corresponding angles are equal and corresponding sides are proportional relative to te scale factor used to dilate. Two different-sized figures can be sown to be similar by using transformations if one of te figures can be mapped onto te oter using a series of transformations, one of wic is a dilation and te oter(s) a reflection, rotation and/or translation. Ex #1: Prove te following figures are similar by describing a series of transformations tat will map Figure 1 onto Figure 2. 2 A. Prove te following figures are similar by describing a series of transformations tat will map te smaller triangle to te larger triangle. B. Are tese triangles similar? Justify your reasoning. One possible solution: Dilate Figure 1 by a scale factor of 2 wit center of dilation at. Ten translate Figure 1 four units rigt and four units down. Ex #2: If ABCD~EFGH, find. Solution: Since te figures are similar, ten a dilation as occurred using a scale factor. Tis creates corresponding sides tat are proportional: BD FG = CD HG 30 6 = CD 16 CD = 80 cm G.SRT.2 G.SRT.2 Page 2 of 11 MCC@WCCUSD 12/11/14

3 3 G.SRT.3-Some Tings To Know Tis standard asks te student to establis tat AA is a similarity criterion for two triangles using a series of transformations. 3 Omar tinks tat if two angles of one triangle are congruent to two angles of anoter triangle, ten te triangles are similar. To sow tis, e drew te figure below. Ex: Pillip draws two triangles. Two pairs of corresponding angles are congruent. Select eac statement tat is true for all suc pairs of triangles. A. A sequence of rigid motions carries one triangle onto te oter. B. A sequence of rigid motions and dilations carries one triangle onto te oter. C. Te two triangles are similar because te triangles satisfy Angle-Angle criterion. D. Te two triangles are congruent because te triangles satisfy Angle-Angle criterion. E. All pairs of corresponding angles are congruent because triangles must ave an angle sum of 180. F. All pairs of corresponding sides are congruent because of te proportionality of corresponding side lengts. Solutions: True 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. F is not true because corresponding sides are not congruent for similar triangles. Wic set of transformations maps ΔABC to ΔDEC and supports Omar s tinking? A. A rotation of 180 clockwise about point C followed by a dilation wit a center of point C and a scale factor of 2. B. A rotation of 180 clockwise point C followed by a dilation wit a center of point C and a scale factor of 1 2. C. A rotation of 180 clockwise point C followed by a dilation wit a center of point C and a scale factor of 3. D. A rotation of 180 clockwise point C followed by a dilation wit a center of point C and a scale factor of 1 3. G.SRT.3 G.SRT.3 Page 3 of 11 MCC@WCCUSD 12/11/14

4 G.SRT.-Some Tings To Know Wen solving problems regarding similarity, remember tat corresponding sides of similar figures are proportional. A. A flagpole 4 meters tall casts a 6-meter sadow.

4 4 G.SRT.-Some Tings To Know Wen solving problems regarding similarity, remember tat corresponding sides of similar figures are proportional. A. A flagpole 4 meters tall casts a 6-meter sadow. At te same time of day, a nearby building casts a 24- meter sadow. How tall is te building? Solution: Draw a picture: 4 A. Mark stands next to a tree tat casts a 1- foot sadow. If Mark is 6 feet tall and casts a 4-foot sadow, ow tall is te tree? 4 m 6 m 24 m Write a proportion and solve: 4 = 24 6 (6)(4) 4 = 24 6 (6)(4) 6 = 24(4) B. Find te value of x in te figure below. 6 6 = 24(4) 6 = (4)(4) =16 Te building is 16 meters tall. B. Find mbe. Let BE = x. Or using Side- Splitting Teorem: G.SRT. G.SRT. Page 4 of 11 MCC@WCCUSD 12/11/14

5 G.SRT.6-Some Tings To Know Similar rigt triangles ave side ratios tat are equal to eac oter. For example, every triangle, no matter wat size, as a small side to ypotenuse ratio of 1:2 or 1 2 (or 0.). Given ΔMAT, matc eac trigonometric ratio to its equivalent value in te box. M 17 T Tese are te side lengt ratio definitions of te acute angle, θ : 1 8 sinθ = Opposite Hypotenuse A cosθ = Adjacent Hypotenuse tanθ = Opposite Adjacent Ex: Write eac trigonometric ratio using te side lengts of ΔABC below. A. B. C. D. E. Solutions: A. B. C. D. E G.SRT.6 G.SRT.6 Page of 11 MCC@WCCUSD 12/11/14

6 6 G.SRT.7-Some Tings To Know 6 Te sine of an angle is equal to te cosine of its complement: sinθ = cos(90 θ). Te table below sows te approximate values of sine and cosine for selected angles. Q A. Fill in te rest of te table. P According to te figure above: sin P = and cos R = R Angle Value of Sine Value of Cosine So, sin P = cos R. If sin ten te cosine of its complement is equivalent: B. Explain ow you determined te values you used. sin32 = cos(90 32 ) = cos Ex: Determine weter te following statements are true or false: A. sin 43 = cos47 B. sin 43 = cos43 C. sin 4 = cos4 D. sin17 = cos(90 17) E. cosθ = sin(90 θ) Solutions: A True, B False, C True, D True, E True G.SRT.7 G.SRT.7 Page 6 of 11 MCC@WCCUSD 12/11/14

7 7 G.SRT.8-Some Tings To Know Drawing and labeling pictures are a great way to solve problems using te trigonometric ratios. Don t forget te Pytagorean Teorem (a! + b! = c! )! A. Te angle of elevation from a landscaped rock to te top of a 30-foot tall flagpole is. Wic of te following equations could be used to find te distance between te rock and te base of te flagpole? Select all tat apply. A. sin7 = 30 x B. cos7 = x 30 C. tan7 = 30 x D. tan33 = x 30 Solution: 7 A. A plane is flying at an elevation of 900 meters. From a point directly underneat te plane, te plane is 1200 meters away from a runway. Select all equations tat can be used to solve for te angle of depression (θ) from te plane to te runway. A. sinθ = 900 B. cosθ = 1200 C. tanθ = D. sinθ = 1200 E. cosθ = 900 F. tanθ = A and B are not correct because te ratios do not correspond to te definitions of te trig ratios. C and D are correct since te tangent opposite ratios of tose angles do sow ypotenuse. Also, ere are some special rigt triangle ratios to memorize: 60 B. Determine te missing side lengts of te following triangles: x y G.SRT.8 End of Study Guide G.SRT.8 Page 7 of 11 MCC@WCCUSD 12/11/14

8 1 You Try Solutions: A. Dilate te following figure using a scale factor of 2 wit center of dilation at te origin. 2 A. Prove te following figures are similar by describing a series of transformations tat will map te smaller triangle to te larger triangle. OR multiply eac vertex s coordinate by te scale factor of 2 to find te 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 dilated ΔABC by a scale factor of 3 wit center of dilation at (2, 1) and ten translated 4 units rigt and 2 units up, ten ΔABC maps onto ΔA'B'C'. B. Dilate te following figure using a scale factor of 1 2 wit center at (4,-2). Anoter solution could be dilating ΔABC by a scale factor of 3 wit center of dilation at te origin. B. Are tese triangles similar? Justify your reasoning. C. Dilate te following figure using a scale factor of -2 wit center at te origin. If te triangles are similar, ten all corresponding side ratios must be equal since a dilation as occurred = = = 3 2 TUV~ WXY. G.SRT.1 G.SRT.2 Page 8 of 11 MCC@WCCUSD 12/11/14

3 Omar tinks tat if two angles of one triangle are congruent to two angles of anoter triangle, ten te triangles are similar. To sow tis, e drew te figure below. 4 A.

6 ft 6 = 1 4 1 ft 4 ft Wic set of transformations maps ΔABC to ΔDEC and supports Omar s tinking? Te scale factor is 2 since te corresponding sides ave a ratio of 6:3, or 2:1.

9 3 Omar tinks tat if two angles of one triangle are congruent to two angles of anoter triangle, ten te triangles are similar. To sow tis, e drew te figure below. 4 A. Mark stands next to a tree tat casts a 1-foot sadow. If Mark is 6 feet tall and casts a 4-foot sadow, ow tall is te tree? 6 ft 6 = ft 4 ft Wic set of transformations maps ΔABC to ΔDEC and supports Omar s tinking? Te scale factor is 2 since te corresponding sides ave a ratio of 6:3, or 2:1. Terefore, A is te correct answer. G.SRT.3 (4)(6) 6 = 1 4 (4)(6) 4 = 90 = 22. Te tree is 22. ft ig. B. Find te value of x in te figure below. (x + 2)+8 8 x (24) x = = (x +10) = 34 3x + 30 = 34 3x = 4 = (24) x = 4 3 G.SRT. Page 9 of 11 MCC@WCCUSD 12/11/14

B C. D D. C E. A F. B 1200 m 900 m G.SRT.6 6 Solution: Use te Pytagorean Teorem to find te lengt of te ypotenuse: Te table below sows te approximate values of sine and cosine for selected angles. a! + b!

10 7 Given ΔMAT, matc eac trigonometric ratio to its equivalent value in te box. M 1 17 A 8 T A. A plane is flying at an elevation of 900 meters. From a point directly underneat te plane, te plane is 1200 meters away from a runway. Select all equations tat can be used to solve for te angle of depression (θ) from te plane to te runway. A. A B. B C. D D. C E. A F. B 1200 m 900 m G.SRT.6 6 Solution: Use te Pytagorean Teorem to find te lengt of te ypotenuse: Te table below sows te approximate values of sine and cosine for selected angles. a! + b! = c! 1200! + 900! = c! A. Fill in te rest of te table = c! Angle Value of Sine Value of Cosine = c! = c B. Based on tis information, te following are equations tat can be used to solve for te angle of depression: C. tanθ = B. Explain ow you determined te values you used. Te sine of an angle is equal to te cosine of its complement. So, sin1 = cos7, sin30 = cos60, sin 4 = cos4 and sin0 = cos90. G.SRT.7 D. sinθ = 1200 E. cosθ = 900 G.SRT.8 Page 10 of 11 MCC@WCCUSD 12/11/14

11 7 Continued B. Determine te missing side lengts of te following triangles: x y Te sorter side s lengt is alf te ypotenuse. So x =11. Te lengt of te longer leg is 3 longer tan te sorter leg, so y = Te altitude of te triangle creates a triangle wit te sorter leg being alf of, wic is 9. Terefore te longer leg is 3 longer tan te sorter leg, so = 9 3. Since two sides are equal, tis is an isosceles triangle, wic means tat te base angles are equal. Eac base angle is 4, so tis is a triangle. Te ypotenuse is 2 longer tan te sides, wic means te side lengts are 7 units. G.SRT.8 Page 11 of 11 MCC@WCCUSD 12/11/14

B. Dilate the following figure using a scale

B. Dilate the following figure using a scale 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