Question

the right triangle below has legs of length $a = 11$ and $b = 8$. 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: part 2: compute the area of the large (outer) square:

Explanation:

Step1: Calculate area of one triangle

The area formula for a right - triangle is \(A=\frac{1}{2}ab\). Given \(a = 11\) and \(b = 8\), for one triangle, \(A_1=\frac{1}{2}\times11\times8=44\).

Step2: Calculate total area of four triangles

Since there are four congruent triangles, \(A_{triangles}=4\times A_1\). Substituting \(A_1 = 44\), we get \(A_{triangles}=4\times44 = 176\).

Step3: Calculate side length of the large square

The side length of the large square \(s=a + b\). Substituting \(a = 11\) and \(b = 8\), \(s=11 + 8=19\).

Step4: Calculate area of the large square

The area formula for a square is \(A=s^2\). Substituting \(s = 19\), \(A_{square}=(11 + 8)^2=19^2=361\).

Answer:

Part 1: \(176\)
Part 2: \(361\)