Question

use a pythagorean triple to find x.
7.
8.
9.
10.

Explanation:

Step1: Recall Pythagorean Triple

A Pythagorean Triple is a set of three positive integers \(a\), \(b\), \(c\) such that \(a^{2}+b^{2}=c^{2}\). Common triples are \((3,4,5)\), \((5,12,13)\), \((8,15,17)\) etc.

Step2: Solve for problem 7

For the triangle with sides \(27\) and \(36\).
We know that \(27 = 9\times3\), \(36=9\times4\). Using the \((3,4,5)\) triple (scaled by \(9\)).
The hypotenuse \(x = 9\times5=45\)

Step3: Solve for problem 8

For the triangle with sides \(120\) and \(136\).
Divide each number by \(8\): \(120\div8 = 15\), \(136\div8=17\). Using the \((8,15,17)\) triple (scaled by \(8\)).
The other leg \(x=8\times8 = 64\)

Step4: Solve for problem 9

For the triangle with sides \(39\) and \(65\).
Divide each number by \(13\): \(39\div13 = 3\), \(65\div13 = 5\). Using the \((3,4,5)\) triple (scaled by \(13\)).
The other leg \(x=13\times4=52\)

Step5: Solve for problem 10

For the triangle with sides \(42\) and \(150\).
Divide each number by \(6\): \(42\div6=7\), \(150\div6 = 25\). Using the \((7,24,25)\) triple (scaled by \(6\)).
The other leg \(x=6\times24 = 144\)

Answer:

1. \(x = 45\)
2. \(x = 64\)
3. \(x = 52\)
4. \(x = 144\)