Question

writing find the length of the hypotenuse of the right triangle. use pencil and paper. explain how you can interpret the pythagorean theorem using the areas of squares. (the figure is not drawn to scale.) the length of the hypotenuse is \\(\square\\) (simplify your answer.)

Explanation:

Step1: Recall Pythagorean Theorem

For a right triangle with legs $a, b$ and hypotenuse $c$, $a^2 + b^2 = c^2$

Step2: Substitute given values

$a=15$, $b=36$, so $15^2 + 36^2 = c^2$
Calculate squares: $225 + 1296 = c^2$

Step3: Sum and solve for c

$1521 = c^2$, so $c = \sqrt{1521}$

Step4: Simplify square root

$\sqrt{1521} = 39$

Step5: Explain area interpretation

The Pythagorean Theorem can be interpreted as: if you draw a square on each side of the right triangle, the area of the square on the hypotenuse (which is $c^2$) is equal to the sum of the areas of the squares on the two legs (which are $a^2$ and $b^2$). For this triangle, the square on the 15-unit leg has area $225$, the square on the 36-unit leg has area $1296$, and the square on the hypotenuse has area $1521$, which is $225+1296$.

Answer:

The length of the hypotenuse is 39.
Interpretation: The area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs of the right triangle.