ANSWER
The length of the diagonal is 5 ft
EXPLANATION
The diagram of the rug with the diagonal is:
The diagonal and the sides form a right triangle, so we can use the Pythagorean theorem to find x, the length of the diagonal:
![\begin{gathered} x^2=3^2+4^2 \\ x=\sqrt[]{9+16} \\ x=\sqrt[]{25} \\ x=5 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%5E2%3D3%5E2%2B4%5E2%20%5C%5C%20x%3D%5Csqrt%5B%5D%7B9%2B16%7D%20%5C%5C%20x%3D%5Csqrt%5B%5D%7B25%7D%20%5C%5C%20x%3D5%20%5Cend%7Bgathered%7D)
A) The answer is 24 m
x - the distance from the spot on the beach to the parking lot
Triangles that are divided by x distance is similar, and therefore:
x : 18 = 32 : x
x * x = 18 * 32
x² = 576
x = √576
x = 24
Thus, the distance <span>the spot on the beach from the parking lot is 24 m.
</span>
b) The answer is 40 m.
x - the distance from the spot on the beach to the parking lot
b - distance from the parking lot to the refreshment stand.
First, find the horizontal leg (a) of the biggest right triangle. According to the Pythagorean theorem:
a² + b² = c²
a = ?
b = ?
c = 32 m + 18 m = 50 m
Find a from the right triangle which sides are a (as a hypotenuse), x, and 18 m.
x² + 18² = a²
x = 24
24² + 18² = a²
576 + 324 = a²
900 = a²
a = √900
a = 30 m
Now, find b:
a² + b² = c²
30² + b² = 50²
900 + b² = 2500
b² = 2500 - 900
b² = 1600
b = √1600
b = 40 m
Thus the distance from the parking lot to get the refreshment stand is 40 m.
