Question

If \(\triangle ABC \sim \triangle ADE\) in the adjoining figure, then which of the following is true?

• A. \(\frac{AB}{BE} = \frac{AC}{CD}\)
• B. \(\frac{AB}{AD} = \frac{AC}{AE}\)
• C. \(\frac{AB}{BC} = \frac{AE}{DE}\)
• D. \(\frac{AC}{AD} = \frac{AB}{AE}\)

Hint:

Always write the ratio based on the order of the letters in the similarity statement. For \(\triangle \textbf{AB}C \sim \triangle \textbf{AD}E\), the first two letters of the first triangle (\(AB\)) always correspond to the first two of the second (\(AD\)).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Similarity between two triangles (\(\triangle ABC \sim \triangle ADE\)) implies that their corresponding angles are equal and their corresponding sides are in the same proportion.
Step 2: Key Formula or Approach:
The ratio of corresponding sides must be equal: \[ \frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE} \]
Step 3: Detailed Explanation:
1. From the similarity \(\triangle ABC \sim \triangle ADE\), match the vertices:
- \(A\) corresponds to \(A\) (Common Angle)
- \(B\) corresponds to \(D\)
- \(C\) corresponds to \(E\)
2. The ratio of the sides is therefore: \(\frac{AB}{AD} = \frac{AC}{AE} = \frac{BC}{DE}\).
3. Comparing this with the options, option (b) matches the proportionality of the corresponding sides.
Step 4: Final Answer:
The correct statement is \(\frac{AB}{AD} = \frac{AC}{AE}\).