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Difficulty: MediumArea of Two-Dimensional Shapes

An artist designs a logo consisting of a rectangle and an isosceles triangle. The rectangle has a length of 1212 centimeters and a width of 88 centimeters. The base of the isosceles triangle is one of the 88-centimeter sides of the rectangle, and the vertex of the triangle lies outside the rectangle. If the total area of the logo is 120120 square centimeters, what is the height, in centimeters, of the triangle?

  1. A
    3
  2. B
    4
  3. 6Answer
  4. D
    12

Answer

6
The total area of the logo is the sum of the area of the rectangle and the area of the triangle. The area of the rectangle is the product of its length and width: 12×8=9612 \times 8 = 96 square centimeters. Subtracting the area of the rectangle from the total area gives the area of the triangle: 120−96=24120 - 96 = 24 square centimeters. The area of a triangle is given by the formula 12×base×height\frac{1}{2} \times \text{base} \times \text{height}. The base of the triangle is one of the 88-centimeter sides of the rectangle. Substituting the base and area into the formula gives 24=12×8×height24 = \frac{1}{2} \times 8 \times \text{height}, which simplifies to 24=4×height24 = 4 \times \text{height}. Dividing both sides by 44 yields a height of 66 centimeters.

Step-by-Step Solution

1
Calculate the area of the rectangle.
The area of the rectangle is 12×8=9612 \times 8 = 96 square centimeters.
The total area of the logo is composite, so we must find the area of the rectangular portion first.
2
Calculate the area of the triangle.
The area of the triangle is 120−96=24120 - 96 = 24 square centimeters.
Subtracting the rectangle's area from the total area of the logo yields the remaining area occupied by the triangle.
3
Solve for the height of the triangle using the area formula.
The height of the triangle is 66 centimeters.
The base of the triangle is 88 centimeters. Using the formula Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}, we set up the equation 24=12×8×height24 = \frac{1}{2} \times 8 \times \text{height}, which simplifies to 24=4×height24 = 4 \times \text{height}, giving a height of 66 centimeters.

Key Concept

The area of a composite shape is the sum of the areas of its simpler component shapes.
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Topics

Area of Two-Dimensional Shapes

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