PSAT Math : How to find if right triangles are similar
Study concepts, example questions & explanations for PSAT Math
All PSAT Math Resources
Example Questions
Example Question #1 : Right Triangles
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
95
85
80
75
90
85
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.
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 #2 : How To Find If Right Triangles Are Similar
Which angle or angles must be complementary to 
I)
II)
III)
IV)
V)
II and V only
I and III only
I only
II only
IV only
II and V only
Since 
The correct response is II and V only.
Example Question #1 : Right Triangles
Refer to the above figure. Given that 
By the Pythagorean Theorem,
The similarity ratio of 
which is subsequently the ratio of the perimeter of 
The perimeter of 
so the perimeter of 
Example Question #4 : How To Find If Right Triangles Are Similar
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
The correct answer is not among the other responses.
By the Pythagorean Theorem,
The similarity ratio of 
This can be used to find 
The area of 
Example Question #451 : Geometry
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that 
By the Pythagorean Theorem, since 
The similarity ratio of 
Likewise,
All PSAT Math Resources
