Triangles and the Pythagorean theorem — SHSAT practice questions

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Practice 93 Triangles and the Pythagorean theorem questions in the app

What this topic is

The Pythagorean theorem says that in a right triangle the squares of the two legs add up to the square of the hypotenuse, the longest side, which lies opposite the right angle.

Recognizing Pythagorean triples, sets of whole-number sides that fit the theorem exactly, saves time. Harder questions give the sides as algebraic expressions, which leads to an equation to solve. Triangles and the Pythagorean theorem also covers angle facts: the angles of any triangle add up to a straight angle, and the base angles of an isosceles triangle are equal.

Some problems combine area with the theorem, using the two legs as base and height.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Harder

A right triangle has legs measuring $x$ and $x + 4$ units and hypotenuse measuring $x + 8$ units. What is the perimeter of the triangle, in units?

44✗ Not this one48✓ Correct52✗ Not this one60✗ Not this one

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Answer: B — 48

$x^{2} + (x + 4)^{2} = (x + 8)^{2}$ simplifies to $x^{2} - 8 x - 48 = 0$ , so $x = 12$ . The side lengths are 12, 16, and 20, and the perimeter is 48.

Question 2Easier

An isosceles triangle has a vertex angle of $4 0^{\circ}$ . What is the measure of each base angle?

4 0^{\circ}

✗ Not this one

5 0^{\circ}

✗ Not this one

7 0^{\circ}

✓ Correct

8 0^{\circ}

✗ Not this one

Show answer

Answer: C — $7 0^{\circ}$

The base angles sum to $180 - 40 = 140$ degrees, so each base angle measures $140 \div 2 = 70$ degrees.

Question 3Harder

A right triangle has a hypotenuse of length 20 units and an area of 96 square units. What is the perimeter of the triangle?

40✗ Not this one44✗ Not this one48✓ Correct52✗ Not this one

Show answer

Answer: C — 48

If the legs are $a$ and $b$ , then $a b = 192$ and $a^{2} + b^{2} = 400$ . Therefore, $(a + b)^{2} = 400 + 2 (192) = 784$ , so $a + b = 28$ and the perimeter is 48.

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Practice 93 Triangles and the Pythagorean theorem questions in the app

Triangles and the Pythagorean theorem — SHSAT practice questions

What this topic is

Sample questions

More SHSAT topics

Mathematics