Exercises: Pythagorean theorem and common triples - Digital SAT Math

A

Recall / Warm-Up

1.

In the right triangle below, which side is the hypotenuse?

A.

Side $a$ (from $B$ to $C$ )

B.

Side $b$ (from $A$ to $C$ )

C.

Side $c$ (from $A$ to $B$ )

D.

The side adjacent to the largest angle

2.

What is the value of $\\sqrt{144}$ ?

3.

A right triangle has legs $a = 3$ and $b = 4$ . Which equation correctly finds the hypotenuse $c$ ?

A.

$c = 3 + 4 = 7$

B.

$c = \\sqrt{3^2 + 4^2} = 5$

C.

$c = 3^2 + 4^2 = 25$

D.

$c = \\sqrt{3 + 4} = \\sqrt{7}$

B

Fluency Practice

1.

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

2.

A right triangle has a hypotenuse of 17 and one leg of length 8. What is the length of the other leg $b$ ?

3.

A right triangle has legs of length 9 and 40. What is the length of the hypotenuse?

4.

A right triangle has a hypotenuse of 10 and one leg of length 6. Recognize the Pythagorean triple involved and find the missing leg.

5.

Which set of side lengths forms a Pythagorean triple (a right triangle with all integer sides)?

A.

$3, 4, 6$

B.

$5, 12, 14$

C.

$8, 15, 17$

D.

$7, 24, 26$

C

Varied Practice

D

Word Problems / Application

1.

A 15-foot ladder leans against a vertical wall. The base of the ladder is placed 9 feet from the wall. How high up the wall does the ladder reach, in feet?

2.

A city park has two straight paths. Path A runs from the fountain at point $F(0, 0)$ to the gazebo at point $G(0, 5)$ . Path B runs from the fountain to the statue at point $S(12, 0)$ . A third path connects the gazebo directly to the statue.

1.

What is the length, in units, of the path from the gazebo $G(0, 5)$ to the statue $S(12, 0)$ ?

2.

Which Pythagorean triple does the triangle $FGS$ represent?

A.

$3\\text{-}4\\text{-}5$

B.

$5\\text{-}12\\text{-}13$

C.

$8\\text{-}15\\text{-}17$

D.

$7\\text{-}24\\text{-}25$

3.

A rectangle has a length of 16 cm and a diagonal of 20 cm. What is the width of the rectangle, in centimeters?

A.

$4$

B.

$12$

C.

$8$

D.

$36$

E

Error Analysis

1.

Alex solved this problem:

"In a right triangle, the two legs have lengths 6 and 8. Find the hypotenuse."

Alex's work:
$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

Alex's answer: $c = 100$

What mistake did Alex make?

A.

Alex added the legs incorrectly.

B.

Alex squared the wrong sides.

C.

Alex forgot to take the square root of 100. The correct answer is $c = \\sqrt{100} = 10$ .

D.

Alex used the wrong formula.

2.

Jordan was asked to determine whether sides 5, 7, and 9 form a right triangle.

Jordan's work:
$5 + 7 = 12$ , and $12 > 9$ , so it is a right triangle.

What error did Jordan make?

A.

Jordan should have used the smallest side as $c$ .

B.

Jordan added the sides instead of using $a^2 + b^2 = c^2$ . The correct test gives $5^2 + 7^2 = 74 \\ne 81 = 9^2$ , so it is NOT a right triangle.

C.

Jordan made an arithmetic error: $5 + 7 \\ne 12$ .

D.

The converse of the Pythagorean theorem cannot be applied here.

F

Challenge / Extension

1.

A right triangle has a hypotenuse of 10 and one leg that is exactly twice the length of the other leg. Find the length of the shorter leg. Express your answer in simplified radical form (e.g., write $2\\sqrt{5}$ ).

2.