Question

for problems 6 - 9, determine which two consecutive whole numbers the length of the hypotenuse is between.

1. $73 = c^{2}$
2. $c^{2}=116$

8.
9.
for problems 10 - 13, determine whether the expression represents a whole number. indicate yes and state the whole number that the expression represents, or indicate no.

1. $sqrt{16}$
2. $sqrt{144}$
3. $sqrt{125}$
4. $sqrt{40}$

Explanation:

Step1: Recall square - root property

For \(c^{2}=a\), \(c = \sqrt{a}\) (we consider the positive root since \(c\) represents the length of a side).

Step2: Find perfect - square bounds for problem 6

We know that \(8^{2}=64\) and \(9^{2}=81\). Since \(64<73<81\), then \(\sqrt{64}<\sqrt{73}<\sqrt{81}\), so \(8 < \sqrt{73}<9\).

Step3: Find perfect - square bounds for problem 7

We know that \(10^{2}=100\) and \(11^{2}=121\). Since \(100<116<121\), then \(\sqrt{100}<\sqrt{116}<\sqrt{121}\), so \(10 < \sqrt{116}<11\).

Step4: Use Pythagorean theorem for problem 8

By the Pythagorean theorem \(c^{2}=6^{2}+10^{2}=36 + 100=136\). We know that \(11^{2}=121\) and \(12^{2}=144\). Since \(121<136<144\), then \(\sqrt{121}<\sqrt{136}<\sqrt{144}\), so \(11 < \sqrt{136}<12\).

Step5: Use Pythagorean theorem for problem 9

By the Pythagorean theorem \(c^{2}=5^{2}+5^{2}=25 + 25=50\). We know that \(7^{2}=49\) and \(8^{2}=64\). Since \(49<50<64\), then \(\sqrt{49}<\sqrt{50}<\sqrt{64}\), so \(7 < \sqrt{50}<8\).

Step6: Evaluate square - roots for problems 10 - 13

• For problem 10: \(\sqrt{16}=4\), so the answer is yes, and the whole number is 4.
• For problem 11: \(\sqrt{144}=12\), so the answer is yes, and the whole number is 12.
• For problem 12: We know that \(11^{2}=121\) and \(12^{2}=144\). Since \(121<125<144\), \(\sqrt{125}\) is not a whole number, so the answer is no.
• For problem 13: We know that \(6^{2}=36\) and \(7^{2}=49\). Since \(36<40<49\), \(\sqrt{40}\) is not a whole number, so the answer is no.

Answer:

1. The length of the hypotenuse \(c\) (where \(c^{2}=73\)) is between 8 and 9.
2. The length of the hypotenuse \(c\) (where \(c^{2}=116\)) is between 10 and 11.
3. The length of the hypotenuse \(c\) (where \(c^{2}=6^{2}+10^{2}=136\)) is between 11 and 12.
4. The length of the hypotenuse \(c\) (where \(c^{2}=5^{2}+5^{2}=50\)) is between 7 and 8.
5. yes, 4
6. yes, 12
7. no
8. no