Question

In the proof of BPT (\(DE \parallel BC\) in \(\triangle ABC\)), since \(\triangle BDE\) and \(\triangle CDE\) have equal areas, the ratio AD/DB is shown to be equal to ________.

• A. DB/AD
• B. AE/EC
• C. EC/AE
• D. AD/AE

Hint:

The statement of BPT is simply: "The ratio of the upper-left segment to the lower-left segment equals the ratio of the upper-right segment to the lower-right segment."
So, \(\frac{AD}{DB} = \frac{AE}{EC}\) is the core result of BPT.

Answer 1

The Correct Option is B

Step 1: Recall the two area ratios set up earlier in the proof.
From the construction, $\frac{\text{ar}(\triangle ADE)}{\text{ar}(\triangle BDE)} = \frac{AD}{DB}$ and $\frac{\text{ar}(\triangle ADE)}{\text{ar}(\triangle CDE)} = \frac{AE}{EC}$, since triangles with the same height have areas proportional to their bases.
Step 2: Bring in the equal-area fact.
Triangles $BDE$ and $CDE$ share the same base $DE$ and lie between the same parallel lines $DE$ and $BC$, so \[ \text{ar}(\triangle BDE) = \text{ar}(\triangle CDE) \]
Step 3: Combine the two results.
Both fractions on the left have the same numerator, $\text{ar}(\triangle ADE)$, and now we know their denominators are equal too. So the two fractions must be equal to each other: \[ \frac{AD}{DB} = \frac{AE}{EC} \]
Step 4: State the conclusion.
So $\frac{AD}{DB}$ is shown to equal $\frac{AE}{EC}$.
\[ \boxed{\dfrac{AE}{EC}} \]