Question

find the length of the third side. if necessary, write in simplest radical form.
4
10
answer attempt 1 out of 2
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Explanation:

Step1: Identify triangle type

It's a right triangle, so use Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is hypotenuse, \(a,b\) are legs. Here, hypotenuse \(c = 10\), one leg \(a = 4\), find the other leg \(b\).

Step2: Apply Pythagorean theorem

Rearrange formula: \(b = \sqrt{c^2 - a^2}\). Substitute \(c = 10\), \(a = 4\):
\(b = \sqrt{10^2 - 4^2} = \sqrt{100 - 16} = \sqrt{84}\).

Step3: Simplify radical

\(\sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2\sqrt{21}\). Wait, no—wait, maybe I mixed up legs and hypotenuse? Wait, the right angle is between the leg of length 4 and the unknown leg, and the hypotenuse is 10? Wait, no—wait, in the diagram, the right angle is at the top, so the vertical leg is 4, the hypotenuse is 10? Wait, no, maybe the hypotenuse is 10, and one leg is 4, so the other leg is \(b\). Wait, no, wait: in a right triangle, hypotenuse is the longest side. So 10 is hypotenuse, 4 is one leg, so the other leg \(b\) is \(\sqrt{10^2 - 4^2} = \sqrt{100 - 16} = \sqrt{84} = 2\sqrt{21}\)? Wait, no, wait, maybe I got the legs wrong. Wait, the diagram: the right angle is at the top, so the two legs are 4 and the unknown, and the hypotenuse is 10. So yes, \(a = 4\), \(c = 10\), so \(b = \sqrt{c^2 - a^2} = \sqrt{100 - 16} = \sqrt{84} = 2\sqrt{21}\)? Wait, no, wait, 84 factors into 421, so \(\sqrt{4*21} = 2\sqrt{21}\). Wait, but maybe I made a mistake. Wait, no—wait, 10 squared is 100, 4 squared is 16, 100 - 16 is 84, square root of 84 is 2sqrt(21). Yes.

Wait, but wait, maybe the hypotenuse is not 10? Wait, the diagram: the side labeled 10 is the hypotenuse? Let me check again. The right angle is at the top, so the two legs are vertical (length 4) and horizontal (unknown), and the hypotenuse is the slant side labeled 10. So yes, hypotenuse is 10, one leg is 4, so the other leg is \(\sqrt{10^2 - 4^2} = \sqrt{84} = 2\sqrt{21}\). Wait, but 21 is 3*7, so can't simplify further. So the length of the third side is \(2\sqrt{21}\)? Wait, no, wait, maybe I mixed up the hypotenuse. Wait, maybe the side labeled 10 is a leg? But 10 is longer than 4, but hypotenuse must be longer than both legs. So 10 is hypotenuse, 4 is a leg, so the other leg is \(\sqrt{10^2 - 4^2} = \sqrt{84} = 2\sqrt{21}\). Yes.

Answer:

\(2\sqrt{21}\)