Question

Triangles O N M and S R Q are shown. Angles O N M and S R Q are congruent. The length of side N M is 10 and the length of side S R is 20. The length of side N O is 8 and the length of side Q R is x.
What value of x will make △ONM similar to △SRQ by the SAS similarity theorem?

16
20
25
50

Answer 1

To determine the value of \( x \) that will make triangles \( \triangle ONM \) and \( \triangle SRQ \) similar by the SAS (Side-Angle-Side) similarity theorem, we need to apply the concepts of proportional sides.

Given that:
- \( \angle ONM \cong \angle SRQ \) (angles are congruent)
- The lengths of the sides are \( NM = 10 \) and \( SR = 20 \)
- The lengths of the sides are \( NO = 8 \) and \( QR = x \)

For the triangles to be similar by the SAS theorem, the ratios of the corresponding sides must be equal:

\[
\frac{NM}{SR} = \frac{NO}{QR}
\]

Substituting the known values:

\[
\frac{10}{20} = \frac{8}{x}
\]

This simplifies to:

\[
\frac{1}{2} = \frac{8}{x}
\]

To cross-multiply, we get:

\[
1 \cdot x = 2 \cdot 8
\]

This results in:

\[
x = 16
\]

Thus, the value of \( x \) that will make \( \triangle ONM \) similar to \( \triangle SRQ \) is

\[
\boxed{16}
\]