Question 123·Easy·Geometry and Trigonometry

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In right triangle $\triangle DEF$ , angle $E$ is the right angle. If the lengths of the sides are $DE = 5$ centimeters and $DF = 13$ centimeters, what is the length of side $EF$ , in centimeters?

When a right triangle problem gives you two sides and a right angle, first identify the hypotenuse as the side opposite the right angle. Use the Pythagorean theorem with the hypotenuse on one side of the equation by itself: if you are missing the hypotenuse, add the squares of the legs; if you are missing a leg, subtract the square of the known leg from the square of the hypotenuse. Take the positive square root at the end, and, when possible, recognize common Pythagorean triples like 3-4-5 and 5-12-13 to get the answer quickly without full computation.

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Spot the hypotenuse

Which side is opposite the right angle at $E$ ? That side must be the hypotenuse and goes on the side of the equation by itself in the Pythagorean theorem.

Set up the correct equation

Once you know which side is the hypotenuse, use $a^2 + b^2 = c^2$ with the legs on the left and the hypotenuse on the right. Which sides are the legs in $\triangle DEF$ ?

Use subtract, not add, when solving for a leg

After you plug in $DE = 5$ and $DF = 13$ , you will get an equation involving $EF^2$ and two numbers. Rearrange it so that $EF^2$ is alone. Do you need to add or subtract to do that?

Desmos Guide

Use Desmos to compute the missing side

Type sqrt(13^2 - 5^2) into Desmos. The value that Desmos returns is the length of $EF$ in centimeters.

Step-by-step Explanation

Identify the hypotenuse and the legs

Since angle $E$ is the right angle, the side opposite angle $E$ is the hypotenuse. That side is $DF$ . The two sides that meet at the right angle, $DE$ and $EF$ , are the legs of the right triangle.

Write the Pythagorean theorem for this triangle

For a right triangle with legs $a$ and $b$ and hypotenuse $c$ , the Pythagorean theorem says

a^2 + b^2 = c^2

Here, $DE$ and $EF$ are the legs, and $DF$ is the hypotenuse, so:

DE^2 + EF^2 = DF^2

Substitute the known side lengths and solve for $EF^2$

You are given $DE = 5$ and $DF = 13$ . Substitute these values into the equation:

5^2 + EF^2 = 13^2 \\[0.7em] 25 + EF^2 = 169

Now isolate $EF^2$ :

EF^2 = 169 - 25 = 144

Find EF and match it to an answer choice

To get $EF$ , take the positive square root (side lengths are positive):

EF = \sqrt{144} = 12

So the length of side $EF$ is 12 centimeters, which corresponds to choice B.

Question 123·Easy·Geometry and Trigonometry

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