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In triangle ABC,D is the midpoint of AB,...

In triangle ABC,D is the midpoint of AB, E is the midpoint of DB and F is the midpoint of BC. If the area of `DeltaABC` is 96, the area of `DeltaAEF` is

A

16

B

24

C

32

D

36

Text Solution

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The correct Answer is:
To find the area of triangle AEF in triangle ABC where D, E, and F are midpoints, we can follow these steps: ### Step 1: Identify the given information We know that: - Triangle ABC has an area of 96. - D is the midpoint of AB. - E is the midpoint of DB. - F is the midpoint of BC. ### Step 2: Use the property of medians Since D is the midpoint of AB, the median AF divides triangle ABC into two triangles of equal area. Therefore, the area of triangle ABF is half of the area of triangle ABC. \[ \text{Area of } \triangle ABF = \frac{1}{2} \times \text{Area of } \triangle ABC = \frac{1}{2} \times 96 = 48 \] ### Step 3: Find the area of triangle ADF Now, we can apply the median property again. Since DF is a median in triangle ABF, it divides triangle ABF into two triangles of equal area. \[ \text{Area of } \triangle ADF = \frac{1}{2} \times \text{Area of } \triangle ABF = \frac{1}{2} \times 48 = 24 \] ### Step 4: Find the area of triangle BFD Next, we need to find the area of triangle BFD. Since F is the midpoint of BC, we can apply the median property again. The area of triangle BFD is half of the area of triangle ABC. \[ \text{Area of } \triangle BFD = \frac{1}{2} \times \text{Area of } \triangle ABC = \frac{1}{2} \times 96 = 48 \] ### Step 5: Find the area of triangle BEF Now, since EF is the median in triangle BFD, it divides triangle BFD into two equal areas. \[ \text{Area of } \triangle BEF = \frac{1}{2} \times \text{Area of } \triangle BFD = \frac{1}{2} \times 48 = 24 \] ### Step 6: Find the area of triangle FED Now we can find the area of triangle FED. Since EF is the median in triangle BFD, we can again apply the median property. \[ \text{Area of } \triangle FED = \frac{1}{2} \times \text{Area of } \triangle BFD = \frac{1}{2} \times 24 = 12 \] ### Step 7: Find the area of triangle AEF Finally, we can find the area of triangle AEF by adding the areas of triangles ADF and FED. \[ \text{Area of } \triangle AEF = \text{Area of } \triangle ADF + \text{Area of } \triangle FED = 24 + 12 = 36 \] ### Conclusion Thus, the area of triangle AEF is **36**. ---
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