Question

solve for x using pythagorean theorem
3
x
4
3²+4²=x²
9+16=x²
25=x²
x=√25=5
solve for x using pythagorean theorem
5
13
x
5²+x²=13²
25+x²=169
solve for x using pythagorean theorem
5
6
x
solve for x using pythagorean theorem
x
10
4
solve for x using pythagorean theorem
10
16
x
solve for x using pythagorean theorem
9
x
5

Explanation:

Step1: Apply Pythagorean theorem to each triangle

For each right triangle, use $a^2 + b^2 = c^2$ where $c$ is hypotenuse.

Triangle 1 (3,4,x):

$3^2 + 4^2 = x^2$ → $9 + 16 = x^2$ → $x^2=25$ → $x=5$

Triangle 2 (5,x,13):

$5^2 + x^2 = 13^2$ → $25 + x^2 = 169$ → $x^2=144$ → $x=12$

Triangle 3 (5,x,6):

$5^2 + x^2 = 6^2$ → $25 + x^2 = 36$ → $x^2=11$ → $x=\sqrt{11}$

Triangle 4 (x,4,10):

$x^2 + 4^2 = 10^2$ → $x^2 + 16 = 100$ → $x^2=84$ → $x=2\sqrt{21}$

Triangle 5 (10,x,16):

$10^2 + x^2 = 16^2$ → $100 + x^2 = 256$ → $x^2=156$ → $x=2\sqrt{39}$

Triangle 6 (9,5,x):

$9^2 + 5^2 = x^2$ → $81 + 25 = x^2$ → $x^2=106$ → $x=\sqrt{106}$

Answer:

5, 12, $\sqrt{11}$, $2\sqrt{21}$, $2\sqrt{39}$, $\sqrt{106}$