Question

In triangle ABC, a line segment DE is parallel to BC, with D on AB and E on AC. If the ratio of the area of triangle ADE to the area of the trapezoid DECB is 4:21, what is the ratio of AD to DB?

• A. 2 : 3
• B. 2 : 5
• C. 3 : 5
• D. 4 : 5

Answer 1

Answer:

A. 2 : 3

Since (DE \parallel BC), triangles (ADE) and (ABC) are similar.

Let:
[
\text{Area}(ADE) : \text{Area}(DECB) = 4 : 21
]

So total area:
[
\text{Area}(ABC) = 4 + 21 = 25 \text{ parts}
]

Thus:
[
\frac{\text{Area}(ADE)}{\text{Area}(ABC)} = \frac{4}{25}
]

---

Step 1: Use similarity ratio

For similar triangles:
[
\frac{\text{Area}(ADE)}{\text{Area}(ABC)} = \left(\frac{AD}{AB}\right)^2
]

So:
[
\left(\frac{AD}{AB}\right)^2 = \frac{4}{25}
]

[
\frac{AD}{AB} = \frac{2}{5}
]

---

Step 2: Find AD : DB

If (AD/AB = 2/5), then:

• (AD = 2x)

• (AB = 5x)

• (DB = AB - AD = 3x)

So:
[
AD : DB = 2x : 3x = 2 : 3
]

---

✅ Final Answer:

[
\boxed{2:3}
]