Question 564·Medium·Math

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A triangular shade sail forms a right triangle. The longest side of the sail is 16 feet, and one of the acute angles measures $45^\circ$ . What is the perimeter of the sail, in feet?

A right triangle with a $45^\circ$ angle is a special right triangle: its legs match, and its hypotenuse is $\sqrt2$ times either leg. Match the known side to its place in the ratio before finding missing sides. For perimeter, add all three sides; adding only one leg stops too early.

Hints

Hint 1
A special right triangle with one $45^\circ$ angle has a second $45^\circ$ angle, so its legs match. The longest side sits across from the right angle. Which side length is given?
Hint 2
The hypotenuse is $\sqrt2$ times either leg in this triangle. Work backward from the known longest side by dividing by $\sqrt2$ . What is the length of each leg?
Hint 3
A triangle’s perimeter is the distance around its entire edge. Add the two equal legs and the longest side; using only one leg leaves out part of the sail’s boundary.

Step-by-step

Use the 45-45-90 side ratio
Step 1
Identify the special right triangle
No Desmos needed. Use the special right triangle ratio. The two angles besides the right angle add to $90^\circ$ , so the other acute angle also measures $45^\circ$ . You have a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle with equal legs.
Step 2
Find the length of each leg
The figure shows $AB=16$ across from the right angle, so $AB$ is the hypotenuse. In this triangle, the hypotenuse is $\sqrt2$ times a leg. Divide the given hypotenuse by $\sqrt2$ to get either leg:
$\text{each leg}=\frac{16}{\sqrt2}$
Multiply top and bottom by $\sqrt2$ , which is multiplying by $1$ :
$\text{each leg}=\frac{16\sqrt2}{2}$
Divide $16$ by $2$ :
$\text{each leg}=8\sqrt2$
Step 3
Add the two legs
The legs are equal, so double one leg to get their combined length:
$2(8\sqrt2)=16\sqrt2$
That counts both legs, but not the longest side.
Step 4
Find the perimeter
A perimeter measures distance around the sail, so all three sides count. Add the hypotenuse to the two-leg total:
$P=16+16\sqrt2$
The sail’s perimeter is $16+16\sqrt2$ feet. Choice B.

Question 564·Medium·Math

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