First, split the figure into 2 shapes: a rectangle and a triangle. You know that the bottom length of the triangle is 4, since the length of the complete figure minus the length of the rectangle is the length of the triangle (11 - 7 = 4). Next, as the hint suggests, use the Pythagorean Theorem to solve for the triangle's side on which it splits from the rectangle:
[tex]a^{2} + b^{2} = c^{2} \\ a^{2} +4^{2} = 5^{2} \\ a^{2} +16=25 \\ a^{2} = 9 \\ a=3[/tex]
*Keep in mind that the square root of 9 is both ±3, but I omitted the negative solution because length is always positive.
The shared side is 3 inches long. Knowing this, we can calculate the area of the rectangle by multiplying length and width:
7 * 3 = 21
To calculate the triangle's height, multiply base times height times one half:
4 * 0.5 * 3 = 6
To find the total area of the figure, add the individual areas of the of the rectangle and triangle:
21 + 6 = 27
The trapezoid's area is 27 in².
