SAT Practice Test - Math - Bisectors and Similarities in Triangles

SAT practice tests arranged by topic and difficulty level. In this section find tips and tactics for solving questions that focus on Bisectors and Similarities in Triangles.

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SAT Practice Test 2015.

In the figure above, AE||CD & and segment AD intersects segment CE at B. What is the length of
segment CE?

Answer:

Angles ABE and DBC have the same measure because they are vertical angles.
Since segment AE is parallel to segment CD, angles A and D are of the same measure (alternate interior angle theorem).
Thus, by the angle-angle theorem, triangle ABE is similar to triangle DBC. Therefore:

$\frac{BC}{BD}=\frac{BE}{AB}$

$(BC)(AB)=(BD)(BE)$
$(BC)(10)=(5)(8)$
$BC=4$

And segment CE measures:
$CE=BC+BE=4+8=12$

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SAT 2015 Test.

In the xy-plane above, line A is parallel to line k. What is the value of p ?
A) 4
B) 5
C) 8
D) 10

Answer:

Since lines "l" and "k" are parallel, they have the same slope:

$\frac{2-0}{0-(-5)}=\frac{0-(-4)}{p-0}$

$\frac{2}{5}=\frac{4}{p}$

$2p=4(5)$
$p=10$

Answer: D

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SAT 2015 Test.

A summer camp counselor wants to find a length, x, in feet, across a lake as represented in the sketch above. The lengths represented by AB, EB, BD, and CD on the sketch were determined to be 1800 feet, 1400 feet, 700 feet, and 800 feet, respectively. Segments AC and DE intersect at B, and ∠AEB and ∠CDB have the same measure. What is the value of x ?

Answer:

∠ABE and ∠CBD have the same measure (vertical angles);
∠AEB and ∠CDB have the same measure (given);
Therefore, ∠EAB and ∠DCB also have the same measure, and triangles EAB and DCB are similar.

Since the triangles are similar, the corresponding sides are in the same proportion; thus:

$\frac{CD}{AE}=\frac{BD}{EB}$

$\frac{800}{x}=\frac{700}{1400}$

$\frac{800}{x}=\frac{1}{2}$

$x=1600$ feet

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SAT Practice Test.

The figure presents line segments A E and B D that intersect at point C. Line segments A B and D E are drawn resulting in two triangles A B C and E D C.

In the preceding figure, triangle A B C is similar to triangle E D C. Which of the following must be true?

A. Line segment A E is parallel to line segment B D.
B. Line segment A E is perpendicular to line segment B D.
C. Line segment A B is parallel to line segment D E.
D. Line segment A B is perpendicular to line segment D E.

Answer:

Given that triangle ABC is similar to triangle EDC, the corresponding angles $A\hat{B}C$ and $E\hat{D}C$ are congruent. If these two angles are congruent, the transversal BD must be crossing two parallel line segments (AB and DE).

The Alternate Interior Angles theorem states that, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Answer: C

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