Question

for problems 1 and 2, find the length of the hypotenuse c. if the answer is not a whole number of units, use square root notation to express the length.
1.
2.

1. the length of the hypotenuse c in problem 2 is between which two consecutive whole numbers? explain.

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\).

Step2: Solve for problem 1

Given \(a = 9\) and \(b = 12\), then \(c^{2}=9^{2}+12^{2}=81 + 144=225\), so \(c=\sqrt{225}=15\).

Step3: Solve for problem 2

Given \(a = 5\) and \(b = 7\), then \(c^{2}=5^{2}+7^{2}=25 + 49 = 74\), so \(c=\sqrt{74}\).

Step4: Find the range for problem 3

We know that \(8^{2}=64\) and \(9^{2}=81\). Since \(64<74<81\), then \(\sqrt{64}<\sqrt{74}<\sqrt{81}\), so \(8 <\sqrt{74}<9\).

Answer:

1. \(c = 15\)
2. \(c=\sqrt{74}\)
3. The length of the hypotenuse \(c\) in problem 2 is between 8 and 9 because \(8^{2}=64<74<81 = 9^{2}\).