PSAT Math : Right Triangles

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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Example Question #123 : Plane Geometry

Three points in the xy-coordinate system form a triangle.

The points are .

What is the perimeter of the triangle?

Possible Answers:

9 + \sqrt{41}

9 + \sqrt{71}

9 + \sqrt{26}

Correct answer:

9 + \sqrt{41}

Explanation:

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is \sqrt{41}.

Example Question #124 : Plane Geometry

Triangle

Based on the information given above, what is the perimeter of triangle ABC?

Possible Answers:

Correct answer:

Explanation:

Triangle-solution

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

Example Question #1 : Right Triangles

Triangles

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

Possible Answers:

80

90

95

85

75

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:

I only

II and V only

IV only

I and III only

II 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 #1 : How To Find If Right Triangles Are Similar

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 #4 : 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, 

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

Possible Answers:

12

9

14.87

13.07

171

Correct answer:

13.07

Explanation:

The Pythagorean Theorem gives us a2 + b2 = c2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so b2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

Example Question #101 : Sat Mathematics

Which of the following could NOT be the lengths of the sides of a right triangle?

Possible Answers:

5, 12, 13

8, 15, 17

12, 16, 20

14, 48, 50

5, 7, 10

Correct answer:

5, 7, 10

Explanation:

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem a2 + b2 = c2

Example Question #102 : Sat Mathematics

Which set of sides could make a right triangle?

Possible Answers:

10, 12, 16

4, 6, 9

6, 7, 8

9, 12, 15

Correct answer:

9, 12, 15

Explanation:

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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