Question

swer the questions below to find how a, b, and c are related.
part 1: compute the total combined area of the four triangles:
part 2: compute the area of the large (outer) 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 $a = 16$ and $b = 5$. 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:

Part 1: Calculate the area of four triangles

The area of a right - triangle is \(A=\frac{1}{2}ab\).
For four such triangles, the total area \(A_{triangles}=4\times\frac{1}{2}ab = 2ab\).

Part 2: Calculate the area of the large (outer) square

The side length of the large square is \(a + b\).
Using the formula for the area of a square \(A = s^{2}\) (where \(s=a + b\)), we get \(A_{outer}=(a + b)^{2}=a^{2}+2ab + b^{2}\).

Part 3: Calculate the area of the small (inner) square

The area of the small square \(A_{inner}=A_{outer}-A_{triangles}\).
Substitute the results from Part 1 and Part 2: \(A_{inner}=(a^{2}+2ab + b^{2})-2ab=a^{2}+b^{2}\). Since \(A_{inner}=c^{2}\), we have \(c^{2}=a^{2}+b^{2}\).

Part 4: Compute \(a^{2}+b^{2}\) when \(a = 16\) and \(b = 5\)

\(a^{2}+b^{2}=16^{2}+5^{2}\).
\(16^{2}=256\) and \(5^{2}=25\).
So \(a^{2}+b^{2}=256 + 25=281\).

Part 5: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

From Part 3, we know that \(a^{2}+b^{2}=c^{2}\).

Answer:

Part 1: \(2ab\)
Part 2: \(a^{2}+2ab + b^{2}\)
Part 3: \(a^{2}+b^{2}\)
Part 4: \(281\)
Part 5: \(=\)