Question

the right triangle below has legs of length $a=8$ and $b=6$. the hypotenuse has length $c$.

answer the questions below to find how $a$, $b$, and $c$ are related.

part 1: compute the total combined area of the four triangles:
$square$
part 2: compute the area of the large (outer) square:
$square$
part 3: using your answers in parts 1 and 2, find the area of the small (inner) square.
$c^{2} = square$
part 4: we are given the side lengths $a=8$ and $b=6$. compute $a^{2}+b^{2}$.
$a^{2}+b^{2} = square$
part 5: use $<$, $>$, or $=$ to complete the statement below.
$a^{2}+b^{2} square c^{2}$

Explanation:

Step1: Calculate area of one triangle

Area of 1 triangle: $\frac{1}{2} \times a \times b = \frac{1}{2} \times 8 \times 6 = 24$

Step2: Find total area of 4 triangles

Total area: $4 \times 24 = 96$

Step3: Find side length of outer square

Side length of outer square: $a + b = 8 + 6 = 14$

Step4: Calculate area of outer square

Area of outer square: $14^2 = 196$

Step5: Find area of inner square

Area of inner square: $196 - 96 = 100$, so $c^2 = 100$

Step6: Compute $a^2 + b^2$

$a^2 + b^2 = 8^2 + 6^2 = 64 + 36 = 100$

Step7: Compare $a^2 + b^2$ and $c^2$

Since both equal 100, they are equal.

Answer:

Part 1: 96
Part 2: 196
Part 3: 100
Part 4: 100
Part 5: $=$