Question

1. 9 yd, \\(\sqrt{115}\\) yd, 11 yd (right triangle with vertical leg 9 yd, horizontal leg \\(\sqrt{115}\\) yd, hypotenuse 11 yd)

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a,b\) are the legs.
Let \(a = 9\), \(b=\sqrt{115}\), \(c = 11\).

Step2: Calculate \(a^2 + b^2\)

\(a^2=9^2 = 81\), \(b^2=(\sqrt{115})^2=115\).
So \(a^2 + b^2=81 + 115=196\).

Step3: Calculate \(c^2\)

\(c^2 = 11^2=121\). Wait, no, wait, maybe I mixed up. Wait, actually, let's check again. Wait, maybe the triangle is right - angled, so let's verify if \(9^2+(\sqrt{115})^2 = 11^2\)? Wait, no, \(9^2 = 81\), \((\sqrt{115})^2=115\), \(81 + 115=196\), and \(11^2 = 121\). Wait, that's not equal. Wait, maybe the hypotenuse is 11? Wait, no, maybe I made a mistake. Wait, actually, maybe the problem is to verify if it's a right triangle. Let's do that.

The Pythagorean theorem states that in a right - triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

First, find the lengths squared:

• For the side of length 9: \(9^2=81\)
• For the side of length \(\sqrt{115}\): \((\sqrt{115})^2 = 115\)
• For the side of length 11: \(11^2=121\)

Now, sum the squares of the two shorter sides: \(81 + 115=196\), and the square of the longest side (11 is not the longest, \(\sqrt{115}\approx10.72\), 9 is 9, 11 is 11. Wait, 11 is longer than \(\sqrt{115}\approx10.72\). So the longest side is 11. Then \(9^2+(\sqrt{115})^2=81 + 115 = 196\), and \(11^2 = 121\). Wait, that's not equal. Wait, maybe there is a miscalculation. Wait, \(14^2=196\), so maybe the hypotenuse is 14? But the diagram says 11. Wait, maybe the problem is to find the missing side? Wait, no, the diagram shows all three sides. Wait, maybe it's a verification problem. Wait, perhaps I misread the side. Wait, the horizontal side is \(\sqrt{115}\), vertical is 9, hypotenuse is 11. Let's recalculate:

\(9^2+(\sqrt{115})^2=81 + 115 = 196=14^2\), but the hypotenuse is given as 11. There is a contradiction. Wait, maybe the problem is to check if it's a right triangle. So according to the Pythagorean theorem, for a right triangle, \(a^{2}+b^{2}=c^{2}\). Let's check:

If we take \(a = 9\), \(b=\sqrt{115}\), then \(a^{2}+b^{2}=81 + 115=196 = 14^{2}\). But the hypotenuse is 11, so \(11^{2}=121
eq196\). So it's not a right triangle? Wait, maybe the diagram is labeled wrong, or maybe I made a mistake. Wait, perhaps the vertical side is 9, horizontal is \(\sqrt{115}\), and hypotenuse is 14? But the problem says 11. Alternatively, maybe the problem is to find the length of the hypotenuse when the legs are 9 and \(\sqrt{115}\). Let's do that.

Using Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 9\), \(b=\sqrt{115}\). Then \(c=\sqrt{9^{2}+(\sqrt{115})^{2}}=\sqrt{81 + 115}=\sqrt{196}=14\). So if the legs are 9 and \(\sqrt{115}\), the hypotenuse should be 14, not 11.

Answer:

If we calculate the hypotenuse for legs 9 and \(\sqrt{115}\), the hypotenuse length is \(\boldsymbol{14}\) yd. (If the problem was to verify, the triangle as labeled is not a right triangle; if to find hypotenuse, it's 14)