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Difficulty: MediumTriangles: Properties, Perimeter, and Area

In triangle ABCABC, the ratio of the side lengths AB:BC:ACAB : BC : AC is 3:4:53 : 4 : 5, and the total area of triangle ABCABC is 2424 square units. A line segment DEDE is drawn parallel to side BCBC, where point DD lies on side ABAB and point EE lies on side ACAC. If the perimeter of triangle ADEADE is exactly half the perimeter of triangle ABCABC, what is the area of trapezoid DBCEDBCE in square units?

  1. A
    6
  2. B
    12
  3. C
    16
  4. 18Answer
  5. E
    20

Answer

The area of trapezoid DBCEDBCE is 1818 square units.
Because line segment DEDE is parallel to side BCBC, triangle ADEADE is similar to triangle ABCABC. Given that the perimeter of triangle ADEADE is half the perimeter of triangle ABCABC, the ratio of their side lengths (the linear scale factor) is 1/21/2. The area ratio of similar triangles is the square of the linear scale factor, which is (1/2)2=1/4(1/2)^2 = 1/4. Thus, the area of triangle ADEADE is 1/4×24=61/4 \times 24 = 6 square units. Subtracting this from the total area gives the area of trapezoid DBCEDBCE: 24−6=1824 - 6 = 18 square units.

Step-by-Step Solution

1
Determine the linear scale factor between triangle ADEADE and triangle ABCABC.
Linear scale factor k=Perimeter(ADE)Perimeter(ABC)=12k = \frac{\text{Perimeter}(ADE)}{\text{Perimeter}(ABC)} = \frac{1}{2}.
Since DE∥BCDE \parallel BC, triangle ADEADE is similar to triangle ABCABC, so the ratio of their perimeters equals the ratio of corresponding side lengths.
2
Calculate the area of triangle ADEADE using the area scale factor k2k^2.
Area(ADE)=k2×Area(ABC)=(12)2×24=14×24=6\text{Area}(ADE) = k^2 \times \text{Area}(ABC) = \left(\frac{1}{2}\right)^2 \times 24 = \frac{1}{4} \times 24 = 6 square units.
The ratio of the areas of two similar figures is the square of their linear scale factor.
3
Subtract the area of triangle ADEADE from the area of triangle ABCABC to find the area of trapezoid DBCEDBCE.
Area(DBCE)=Area(ABC)−Area(ADE)=24−6=18\text{Area}(DBCE) = \text{Area}(ABC) - \text{Area}(ADE) = 24 - 6 = 18 square units.
Trapezoid DBCEDBCE is formed by removing triangle ADEADE from triangle ABCABC.

Key Concept

Properties of Similar Triangles and Area Scaling
Estimated Time:1m 30s
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Topics

Triangles: Properties, Perimeter, and Area

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