Question

find the length of the third side. if necessary, round to the nearest tenth. 9 13

Explanation:

Step1: Identify triangle type

It's a right - triangle, so use Pythagorean theorem: \(a^{2}+b^{2}=c^{2}\), where \(c\) is hypotenuse, \(a = 9\), \(b = 13\)? Wait, no, wait. Wait, in a right - triangle, the hypotenuse is the longest side. Wait, maybe I mixed up. Wait, the two legs are 9 and 13? No, wait, the right angle is between the two sides of length 9 and 13? Wait, no, the hypotenuse is the side opposite the right angle. Wait, let's check: if the two legs are \(a = 9\) and \(b = 13\), then the hypotenuse \(c=\sqrt{a^{2}+b^{2}}\). Wait, no, wait, maybe the hypotenuse is one of them? Wait, no, 13 is longer than 9, so if the right angle is between 9 and the unknown side, then 13 is the hypotenuse. Wait, let's clarify. Let's assume the right triangle has legs \(x\) and 9, and hypotenuse 13. Then by Pythagorean theorem: \(x^{2}+9^{2}=13^{2}\). Wait, or if 13 and 9 are legs, then hypotenuse is \(\sqrt{9^{2}+13^{2}}\). Wait, the diagram: the right angle is between the side of length 9 and the unknown side, and the side of length 13 is the hypotenuse? Wait, no, looking at the triangle, the right angle is between the two sides: one is 9, one is the unknown, and the hypotenuse is 13? Wait, no, 13 is longer than 9, so if the right angle is between 9 and the other leg, then hypotenuse is 13. So let's define: let the unknown side be \(x\), legs are \(x\) and 9, hypotenuse is 13. Then \(x^{2}+9^{2}=13^{2}\). Wait, no, \(x^{2}=13^{2}-9^{2}\). Wait, \(13^{2}=169\), \(9^{2}=81\), so \(x^{2}=169 - 81=88\), so \(x=\sqrt{88}\approx9.4\)? Wait, no, wait, maybe I got the legs wrong. Wait, maybe 9 and 13 are the legs, and the hypotenuse is \(c\). Then \(c=\sqrt{9^{2}+13^{2}}=\sqrt{81 + 169}=\sqrt{250}\approx15.8\). Wait, the diagram: the triangle has a right angle, one side is 9, one is 13, and the third is unknown. Let's check the lengths. If 9 and 13 are legs, then hypotenuse is \(\sqrt{9^{2}+13^{2}}=\sqrt{81 + 169}=\sqrt{250}\approx15.8\). If 13 is hypotenuse and 9 is a leg, then the other leg is \(\sqrt{13^{2}-9^{2}}=\sqrt{169 - 81}=\sqrt{88}\approx9.4\). Now, looking at the triangle, the side of length 13 is longer than 9, so if the right angle is between 9 and the unknown side, then 13 is hypotenuse. But maybe the right angle is between 9 and 13? Wait, the diagram: the right angle is marked between the side of length 9 and the side of length 13? No, the right angle symbol is between the side of length 9 and the unknown side. Wait, the triangle is drawn with the right angle at the vertex where the side of length 9 and the side of length 13 meet? No, the right angle symbol is at the vertex where the side of length 9 and the unknown side meet. Wait, maybe I misread. Let's re - examine: the triangle has three sides: one is 9 (adjacent to right angle), one is 13 (adjacent to right angle), and the hypotenuse is unknown. Oh! That's the mistake. The right angle is between the two sides of length 9 and 13, so they are the legs, and the hypotenuse is the unknown side. So that's the key. So legs \(a = 9\), \(b = 13\), hypotenuse \(c\). Then by Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).

Step2: Apply Pythagorean theorem

Calculate \(a^{2}+b^{2}\): \(9^{2}=81\), \(13^{2}=169\), so \(a^{2}+b^{2}=81 + 169 = 250\). Then \(c=\sqrt{250}\). Simplify \(\sqrt{250}=\sqrt{25\times10}=5\sqrt{10}\approx5\times3.1623\approx15.8\).

Answer:

\(15.8\)