Question

Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units

Answer 1

20

Explanation:

We use the Pythagorean Theorem to verify that triangle $ABC$ is a right triangle, or we recognize that $(6,8,10)$ is a multiple of the Pythagorean triple $(3,4,5)$. The area of a right triangle is $\frac{1}{2}bh$ where $b$ and $h$ are the lengths of the two legs, so the area of triangle $ABC$ is $\frac{1}{2}(6)(8)=24$. If the area of the rectangle is $24$ square units and the width is $4$ units, then the length is $\frac{24}{4}=6$ units. That makes the perimeter $6+6+4+4=\boxed{20}$ units.

Answer 2

20

Explanation:

Given that the area of the rectangle is equal to that of the triangle

Area of triangle $ABC$

= 1/2 (bh)

Given that the sides of the triangle are $6$ units, $8$ units, and $10$ units,

The base and the heights are $6$ units and $8$ units. The $10$ units is the hypotenuse

From Pythagoras theorem,

6^2 + 8^2 = 10^2

Therefore, area of triangle

=1/2 (6 × 8)

= $24$ units^2

Area of rectangle = L × W

Where L = Length, W = Width

Area of the rectangle = area of triangle

L × 4 = 24

L= 24/4

L = $6$ Units

Perimeter of rectangle

=2 (L + B)

= 2(6 + 4)

= $20$ Units