Question 588·Medium·Math

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A rectangular community garden has a straight gravel path connecting one corner of the garden to the opposite corner. The path is $24$ meters long. At one end of the path, the angle between the path and one side of the garden is $60^\circ$ .

Which choice is the area of the garden, in square meters?

A rectangle’s diagonal and two sides form a right triangle. A marked $60^\circ$ angle makes it a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, so match the diagonal to the hypotenuse in the exact side ratio. Find both rectangle sides, then multiply for area. Finding one triangle’s area gives only half the garden’s area.

Hints

Hint 1
The rectangle’s sides meet at a right angle. With the marked $60^\circ$ angle, what is the third angle? Which side is the hypotenuse, the side across from the right angle?
Hint 2
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg, long leg, and hypotenuse have lengths $x$ , $x\sqrt3$ , and $2x$ . The diagonal is the hypotenuse. What does its length make $x$ ?
Hint 3
A rectangle’s area is the product of its two sides that meet at a right angle. Multiply the two legs you found, not the diagonal. Halving that product would give the area of only one triangle.

Step-by-step

Use the special right triangle
Step 1
Recognize the triangle
No Desmos needed. The special right triangle ratio gives exact side lengths. The sides of the rectangle meet at a right angle at $B$ . Angle $A$ is $60^\circ$ , so angle $C$ is $30^\circ$ because a triangle’s angles total $180^\circ$ . The path $AC$ is the hypotenuse, the side across from the right angle.
Step 2
Find the short side
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg is across from $30^\circ$ , the long leg is across from $60^\circ$ , and their lengths and the hypotenuse are $x:x\sqrt3:2x$ . Match each side to the angle across from it before using the ratio. So $AB$ is the short leg, while the $24$ -meter path is $2x$ . Match the path to $2x$ :
$2x=24$
Divide by $2$ :
$x=12$
So $AB=12$ meters.
Step 3
Find the long side
Side $BC$ is across from $60^\circ$ , so it’s the long leg. Use $x=12$ :
$BC=x\sqrt3=12\sqrt3$
Don’t call $AB$ the long leg because it touches the $60^\circ$ angle; the long leg is across from that angle.
Step 4
Find the garden’s area
The garden is the whole rectangle, not one of the triangles. Area is the space inside it, so multiply its perpendicular sides:
$A=AB\cdot BC=(12)(12\sqrt3)=144\sqrt3$
The garden’s area is $144\sqrt3$ square meters. Choice B.

Question 588·Medium·Math

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