Similar Triangles

Question 1

(a)  

Answer explanation

The correct answer is 1600. It is given that ∠AEB and ∠CDB have the same measure. Since ∠ABE and ∠CBD are vertical angles, they have the same measure. Therefore, triangle EAB is similar to triangle DCB because the triangles have two pairs of congruent corresponding angles (angle-angle criterion for similarity of triangles). Since the triangles are similar, the corresponding sides are in the same proportion; thus, CD/x=BD/EB. Substituting the given values of 800 for CD, 700 for BD, and 1400 for EB in CD/x=BD/EB gives 800/x=700/1400. Plug that into Desmos. The vertical line will be x=1600.

Tags

CCSS.HSG.CO.C.9

Question 2

(a)  

Answer explanation

The correct answer is 12. Angles ABE and DBC are vertical angles and thus have the same measure. Since segment AE is parallel to segment CD, angles A and D are of the same measure by the alternate interior angle theorem. Thus, by the angle-angle theorem, triangle ABE is similar to triangle DBC, with vertices A, B, and E corresponding to vertices D, B, and C, respectively. Thus, AB/DB=EB/CB or 10/5=8/x. Plug that into desmos and you get x=CB=4. Then CE=CB+BE=4+8=12.

Tags

CCSS.HSG.CO.C.10

Question 3

(a)  

Answer explanation

The correct answer is 20. By the equality given, the three pairs of corresponding sides of the two triangles are in the same proportion. By the side-side-side (SSS) similarity theorem, triangle ABC is similar to triangle DEF. In similar triangles, the measures of corresponding angles are congruent. Since angle BAC corresponds to angle EDF, these two angles are congruent and their measures are equal. It’s given that the measure of angle BAC is 20°, so the measure of angle EDF is also 20°.

Tags

CCSS.HSG.SRT.A.2

Question 4

• A
• B
• C
• D

Answer explanation

Tags

CCSS.HSG.CO.C.10

• A

• B

• C

• D

Answer explanation

Choice B is correct. It’s given that angle ACE measures 62°. Since segments AE and BD are parallel, angles BDC and CEA are congruent. Therefore, angle CEA measures 58°. The sum of the measures of angles ACE, CEA, and CAE is 180° since the sum of the interior angles of triangle ACE is equal to 180°. Let the measure of angle CAE be x°. Therefore, 62+58+x=180, which simplifies to x = 60. Thus, the measure of angle CAE is 60°.

Choice A is incorrect. This is the measure of angle AEC, not that of angle CAE. Choice C is incorrect. This is the measure of angle ACE, not that of CAE. Choice D is incorrect. This is the sum of the measures of angles ACE and CEA.

Tags

CCSS.HSG.SRT.B.4

• A

• B

• C

• D

Answer explanation

Choice D is correct. It's given that triangle ABC is similar to triangle XYZ, such that A, B, and C correspond to X, Y, and Z, respectively. Therefore, side AB corresponds to side XY. Since the length of each side of triangle XYZ is 2 times the length of its corresponding side in triangle ABC, it follows that the measure of side XY is 2 times the measure of side AB. Thus, since the measure of side AB is 16, then the measure of side XY is 2(16), or 32.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the measure of side AB, not side XY.

Choice C is incorrect and may result from conceptual or calculation errors.

Tags

CCSS.7.G.A.1

• A

• B

• C

• D

Answer explanation

Choice C is correct. It’s given that triangle XYZ is similar to triangle RST, such that X, Y, and Z correspond to R, S, and T, respectively. Since corresponding angles of similar triangles are congruent, it follows that the measure of ∠Z is congruent to the measure of ∠T. It’s given that the measure of ∠Z is 20°. Therefore, the measure of ∠T is 20°.

Choice A is incorrect and may result from a conceptual error.

Choice B is incorrect. This is half the measure of ∠Z.

Choice D is incorrect. This is twice the measure of ∠Z.

Tags

CCSS.HSG.SRT.A.2

(a)  

Answer explanation

The correct answer is either 10/3, 15/4, or 25/6. The area of triangle ABC can be expressed as (1/2)(5+7)y or 6y. It’s given that the area of triangle ABC is at least 48 but no more than 60. It follows that 48 ≤ 6y ≤ 60. Dividing by 6 to isolate y in this compound inequality yields 8 ≤ y ≤ 10. Since y is an integer, y = 8, 9, or 10. In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle A. Therefore, the following proportion is true: x/y = 5/12.

Substituting 8 for y in the proportion results in x/8 = 5/12. Plug into desmos and you get x = 10/3.

Substituting 9 for y in the proportion results in x/9 = 5/12. Plug into desmos and you get x = 15/4.

Substituting 10 for y in the proportion results in x/10 = 5/12. Plug into desmos and you get x = 25/6

Tags

CCSS.6.G.A.1

(a)  

Answer explanation

The correct answer is 30. In the figure given, since BD is parallel to AE and both segments are intersected by CE, then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, BD corresponds to AE and CD corresponds to CE. Therefore, BD/CD = AE/CE. Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: 6^2+8^2=CD^2. Taking the square root of each side gives CD = 10. Substituting the values in the proportion BD/CD = AE/CE yields 6/10=18/CE. Plug into desmos using x instead of CE and you get x = CE = 30.

Tags

CCSS.HSG.SRT.B.4

• A

• B

• C

• D

Answer explanation

Choice B is correct. Since figures are drawn to scale unless otherwise stated and triangle ABC is similar to triangle DEF, it follows that the measure of angle B is equal to the measure of angle E. Furthermore, it follows that side AB corresponds to side DE and that side BC corresponds to side EF. For similar triangles, corresponding sides are proportional, so AB/BC = DE/EF. In right triangle DEF, the cosine of angle E, or cos(E), is equal to the length of the side adjacent to angle E divided by the length of the hypotenuse in triangle DEF. Therefore, cos(E) = DE/EF, which is equivalent to AB/BC. Therefore, cos(E) = 12/13.

Choice A is incorrect. This value represents the tangent of angle F, or tan(F), which is defined as the length of the side opposite angle F divided by the length of the side adjacent to angle F.

Choice C is incorrect. This value represents the tangent of angle E, or tan(E), which is defined as the length of the side opposite angle E divided by the length of the side adjacent to angle E.

Choice D is incorrect. This value represents the sine of angle E, or sin(E), which is defined as the length of the side opposite angle E divided by the length of the hypotenuse.

Tags

CCSS.HSG.SRT.A.2