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The area of triangleABC is 88cm^2. If D ...

The area of `triangleABC` is `88cm^2`. If D is the midpoint of BC and E is the midpoint of AB, then the area (in `cm^2`) of `triangleBDE` is:

A

44

B

22

C

11

D

5.5

Text Solution

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The correct Answer is:
To find the area of triangle BDE given that the area of triangle ABC is 88 cm², we can follow these steps: ### Step 1: Understand the Midpoints Let D be the midpoint of side BC and E be the midpoint of side AB in triangle ABC. This means that BD = DC and AE = EB. ### Step 2: Use the Properties of Similar Triangles Since D and E are midpoints, triangle BDE is similar to triangle ABC. The ratio of their corresponding sides will be 1:2 because the lengths of the segments created by the midpoints are half of the lengths of the sides of triangle ABC. ### Step 3: Calculate the Area Ratio The area of similar triangles is proportional to the square of the ratio of their corresponding sides. Therefore, if the ratio of the sides is 1:2, the ratio of the areas will be: \[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] ### Step 4: Apply the Area Ratio to Find Area of Triangle BDE Let the area of triangle BDE be A_BDE. Then, using the area ratio: \[ \frac{A_{BDE}}{A_{ABC}} = \frac{1}{4} \] Given that the area of triangle ABC is 88 cm², we can set up the equation: \[ \frac{A_{BDE}}{88} = \frac{1}{4} \] ### Step 5: Solve for A_BDE To find A_BDE, multiply both sides by 88: \[ A_{BDE} = 88 \times \frac{1}{4} = 22 \text{ cm}^2 \] ### Conclusion Thus, the area of triangle BDE is 22 cm². ---
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