Question

apply the 45°-45°-90° triangle theorem to find the length of a leg of a right triangle if the length of the hypotenuse is 10 cm. 7 cm \sqrt{10\sqrt{2}} cm 4 cm 6 cm 6. choose the correct answer. apply the 45°-45°-90° triangle theorem to find the length of the hypotenuse of a right triangle if the length of a leg is 7 in. \sqrt{\sqrt{7}\sqrt{2}} in 5 in 10 in 4 in

Explanation:

First Sub - Question (Finding leg length with hypotenuse 10 cm)

Step1: Recall 45 - 45 - 90 Triangle Theorem

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the legs are equal and the hypotenuse \(h\) is related to the leg length \(l\) by the formula \(h = l\sqrt{2}\).

Step2: Solve for leg length \(l\)

We know \(h = 10\space cm\) and \(h=l\sqrt{2}\). So, we can solve for \(l\) by rearranging the formula: \(l=\frac{h}{\sqrt{2}}\). Substituting \(h = 10\) into the formula, we get \(l=\frac{10}{\sqrt{2}}\). Rationalizing the denominator (multiplying numerator and denominator by \(\sqrt{2}\)), we have \(l=\frac{10\sqrt{2}}{2}=5\sqrt{2}\space cm\)? Wait, no, wait. Wait, the options given: Wait, maybe I made a mistake. Wait, the options have \(\sqrt{10\sqrt{2}}\)? No, wait, maybe the options are mis - written? Wait, no, let's re - check. Wait, in a \(45 - 45 - 90\) triangle, leg \(l\), hypotenuse \(h = l\sqrt{2}\). So if \(h = 10\), then \(l=\frac{10}{\sqrt{2}}=\frac{10\sqrt{2}}{2} = 5\sqrt{2}\approx7.07\space cm\). But the options have 7 cm, \(\sqrt{10\sqrt{2}}\) cm, 4 cm, 6 cm. Maybe the intended answer is 7 cm (approximate) or maybe there's a mistake in the options. But according to the theorem, the exact value is \(5\sqrt{2}\approx7.07\), so the closest is 7 cm. Wait, but let's check the second sub - question.

Second Sub - Question (Finding hypotenuse with leg 7 in)

Step1: Recall 45 - 45 - 90 Triangle Theorem

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the hypotenuse \(h\) is related to the leg length \(l\) by the formula \(h = l\sqrt{2}\).

Step2: Substitute the leg length

We know that the leg length \(l = 7\space in\). Substituting into the formula \(h=l\sqrt{2}\), we get \(h = 7\sqrt{2}\space in\). But the options have \(\sqrt{7\sqrt{2}}\) in, 5 in, 10 in, 4 in. Wait, maybe there are errors in the options. But if we consider the first sub - question, the closest to \(5\sqrt{2}\approx7.07\) is 7 cm. For the second sub - question, the hypotenuse should be \(7\sqrt{2}\approx9.899\), which is close to 10 in.

Answer:

(First Sub - Question): 7 cm