PSAT Math : How to find if right triangles are similar

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #91 : Triangles

Triangles

In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?

Possible Answers:

80

95

75

90

85

Correct answer:

85

Explanation:

Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.

Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.

We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.

We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:

 a2 + b2 = c2

152 + 202 = c2

625 = c2

c = 25

The length of BD is 25.

Similar_triangles

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of AB, BC, CD, and DA.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85. 

Example Question #1 : Right Triangles

 and  is a right angle.

Which angle or angles must be complementary to  ?

I) 

II) 

III) 

IV) 

V) 

Possible Answers:

IV only

I only

I and III only

II only

II and V only

Correct answer:

II and V only

Explanation:

 is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.

 can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.

Since  is right angle,  is a right triangle, and  and  are its acute angles. That makes  complementary to . Since  is congruent to , it is also complementary to .

The correct response is II and V only.

Example Question #2 : Right Triangles

Triangles

Refer to the above figure. Given that , give the perimeter of .

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem, 

 

The similarity ratio of  to  is 

which is subsequently the ratio of the perimeter of   to that of .

The perimeter of  is 

,

so the perimeter of  can be found using this ratio:

Example Question #5 : Right Triangles

Triangles

Note: Figure NOT drawn to scale.

Refer to the above figure. Given that , give the area of .

Possible Answers:

The correct answer is not among the other responses.

Correct answer:

Explanation:

By the Pythagorean Theorem, 

 

The similarity ratio of  to  is 

 

This can be used to find  : 

 

The area of  is therefore 

Example Question #3 : Right Triangles

Triangles

Note: Figures NOT drawn to scale.

Refer to the above figure. Given that  , evaluate .

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem, since   is the hypotenuse of a right triangle with legs 6 and 8, its measure is 

.

The similarity ratio of  to  is

.

Likewise, 

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