Question

in the diagram below, \\(\overline{op}\\) is parallel to \\(\overline{lm}\\). if \\(nm = 10.5\\), \\(on = 10\\), and \\(ln = 17.5\\), find the length of \\(\overline{np}\\). figures are not necessarily drawn to scale.

diagram of a triangle with right angles at p and m, points n, o, l, p, m as labeled

answer attempt 1 out of 2

\\(np = \square\\) submit answer

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{OP} \parallel \overline{LM} \), \( \triangle NOP \sim \triangle NLM \) by the Basic Proportionality Theorem (AA similarity, as both triangles share \( \angle N \) and have right angles).

Step2: Set Up Proportion

Let \( NP = x \). Then \( NL = ON + OL \), but actually, using the similarity ratio: \( \frac{NP}{NM} = \frac{ON}{LN} \)? Wait, no. Wait, \( LN = 17.5 \), \( ON = 10 \), so \( OL = LN - ON = 17.5 - 10 = 7.5 \)? Wait, no, \( LN \) is the length from \( L \) to \( N \), and \( ON \) is from \( O \) to \( N \). So the sides: \( \frac{NP}{NM} = \frac{ON}{LN} \)? Wait, no, similarity ratio: corresponding sides. \( \triangle NOP \sim \triangle NLM \), so \( \frac{NP}{NM} = \frac{ON}{LN} \)? Wait, \( LN = 17.5 \), \( ON = 10 \), \( NM = 10.5 \). Wait, let's correct: \( LN = ON + OL \)? No, \( N \) is the vertex, \( O \) is on \( NL \), \( P \) is on \( NM \). So \( NL = 17.5 \), \( ON = 10 \), so \( OL = NL - ON = 17.5 - 10 = 7.5 \). Then, since \( \triangle NOP \sim \triangle NLM \), the ratio of sides is \( \frac{ON}{LN} = \frac{NP}{NM} \). Wait, \( ON = 10 \), \( LN = 17.5 \), \( NM = 10.5 \). So:

\( \frac{NP}{10.5} = \frac{10}{17.5} \)

Step3: Solve for \( NP \)

Cross - multiply: \( NP\times17.5 = 10\times10.5 \)

\( NP=\frac{10\times10.5}{17.5} \)

Calculate numerator: \( 10\times10.5 = 105 \)

Then \( NP=\frac{105}{17.5}=6 \)

Answer:

\( 6 \)