Question

If AM and PN are the altitudes of two similar triangles \(∆ABC\) and \(∆PQR\) respectively and \((AB)^ 2 : (PQ) ^2 = 4 : 9\), then \(AM : PN =\)

• A. 3 : 2
• B. 16 : 81
• C. 4 : 9
• D. 2 : 3

Answer 1

The Correct Option is D

Given: \( \triangle ABC \sim \triangle PQR \), and the ratio of squared sides is:

\[ (AB)^2 : (PQ)^2 = 4 : 9 \]

Step 1: Understanding similarity property 

In similar triangles, corresponding altitudes are proportional to their respective sides.

\[ \frac{AM}{PN} = \frac{AB}{PQ} \]

Step 2: Solve for \( AM : PN \)

\[ \left(\frac{AB}{PQ}\right)^2 = \frac{4}{9} \] \[ \frac{AB}{PQ} = \sqrt{\frac{4}{9}} = \frac{2}{3} \] \[ AM : PN = 2 : 3 \]

Final Answer: 2 : 3

Answer 2

To solve this problem, we'll utilize the properties of similar triangles. If two triangles are similar, the ratios of their corresponding sides and corresponding altitudes are equal. Given:

The triangles \(∆ABC\) and \(∆PQR\) are similar. Let AM and PN be the altitudes of these triangles. We know the ratio of the squares of corresponding sides:

\( (AB)^2 : (PQ)^2 = 4 : 9 \).

To find AM : PN, we use the property that the altitudes of similar triangles are proportional to the corresponding sides. Thus:

\( (AM/PN) = (AB/PQ) \).

From the given, we have:

\( (AB)^2 : (PQ)^2 = 4 : 9 \), therefore:

\( (AB/PQ)^2 = 4/9 \).

By taking the square root of both sides, we find:

\( AB/PQ = \sqrt{4/9} = 2/3 \).

Thus, the ratio of the altitudes is:

\( AM : PN = AB : PQ = 2 : 3 \).

Therefore, the correct answer is 2 : 3.