A right triangle has one leg that measures centimeters and a hypotenuse that measures centimeters. What is the length of the triangle's other leg, in centimeters?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 7 | Free SAT Prep | Aniko

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Question 7·Easy·Right Triangles and Trigonometry

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A right triangle has one leg that measures $6$ centimeters and a hypotenuse that measures $10$ centimeters. What is the length of the triangle's other leg, in centimeters?

When you know the hypotenuse and one leg of a right triangle, quickly apply the Pythagorean theorem $a^2 + b^2 = c^2$ , making sure the largest given side is $c$ . Rearrange to $\text{(missing leg)}^2 = c^2 - \text{(known leg)}^2$ , compute the difference of squares, and then take the positive square root. This avoids unnecessary steps and helps you spot and rule out answer choices that come from incorrectly adding squares or just adding side lengths.

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Recall the key right triangle relationship

For a right triangle with legs $a$ and $b$ and hypotenuse $c$ , think about the formula that relates $a$ , $b$ , and $c$ using squares.

Set up the equation correctly

Label the unknown leg as $x$ . Which side is the hypotenuse (the longest side)? Make sure that side is the one that equals $c$ in your formula.

Solve for the unknown leg

After plugging in $6$ and $10$ , isolate $x^2$ by moving the known square to the other side. Then take the square root at the end to get $x$ .

Desmos Guide

Use Desmos to compute the missing leg directly

In Desmos, type the expression sqrt(10^2 - 6^2) to represent the length of the unknown leg using the Pythagorean theorem, and read the numerical value of this expression from the calculator.

Step-by-step Explanation

Identify the known sides and the formula

In a right triangle, the Pythagorean theorem says

a^2 + b^2 = c^2

where $c$ is the hypotenuse (the side opposite the right angle and the longest side). Here, the hypotenuse is $10$ cm, and one leg is $6$ cm. Let the unknown leg be $x$ . So we have:

6^2 + x^2 = 10^2

Plug in and simplify the squares

Compute the squares:

$6^2 = 36$
$10^2 = 100$

So the equation becomes:

36 + x^2 = 100

Isolate the unknown squared term

Subtract $36$ from both sides to isolate $x^2$ :

\begin{aligned} 36 + x^2 &= 100 \\ x^2 &= 100 - 36 \\ x^2 &= 64 \end{aligned}

Find the length of the missing leg and match to the choices

Take the positive square root (side lengths are positive):

x = \sqrt{64}

So the other leg is $8$ centimeters, which corresponds to answer choice B.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation