Question

a. verify that each set of numbers forms a pythagorean triple. 8, 15, 17 and 16, 30, 34
b. write one more pythagorean triple by multiplying each number in the 8, 15, 17 triple by 4.
c. draw a right - triangle for each of the pythagorean triples in parts a and b on the grid. for each triangle, put the right angle at the origin and the shorter leg along the positive x - axis.
d. what do you notice about the three triangles?
a. the set of numbers 8, 15, 17 a dropdown with options is and is not a pythagorean triple because a dropdown with some options. the set of numbers 16, 30, 34 a dropdown with options is and is not a pythagorean triple because a dropdown with some options.

Explanation:

Step1: Recall Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with side lengths \(a\), \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Check the triple 8, 15, 17

For \(a = 8\), \(b = 15\), and \(c = 17\), calculate \(a^{2}+b^{2}\) and \(c^{2}\).
\[a^{2}+b^{2}=8^{2}+15^{2}=64 + 225=289\]
\[c^{2}=17^{2}=289\]
Since \(8^{2}+15^{2}=17^{2}\), the set of numbers 8, 15, 17 is a Pythagorean triple.

Step3: Check the triple 16, 30, 34

For \(a = 16\), \(b = 30\), and \(c = 34\), calculate \(a^{2}+b^{2}\) and \(c^{2}\).
\[a^{2}+b^{2}=16^{2}+30^{2}=256+900 = 1156\]
\[c^{2}=34^{2}=1156\]
Since \(16^{2}+30^{2}=34^{2}\), the set of numbers 16, 30, 34 is a Pythagorean triple.

Answer:

The set of numbers 8, 15, 17 is a Pythagorean triple because \(8^{2}+15^{2}=64 + 225=289=17^{2}\).
The set of numbers 16, 30, 34 is a Pythagorean triple because \(16^{2}+30^{2}=256 + 900=1156=34^{2}\).