Question

solve for x. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.
triangle mlk with side ml = 11, side mk = 16, angle at m is 59°, angle at l is 79°, angle at k is 42°.
triangle opn with side pn = 46, angle at p is 59°, angle at o is 79°, angle at n is 42°, side op is x.
answer attempt 1 out of 2
x = blank
submit answer

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle MKL \) and \( \triangle PNO \) have the same angle measures: \( \angle M = \angle P = 59^\circ \), \( \angle L = \angle O = 79^\circ \), \( \angle K = \angle N = 42^\circ \). So, they are similar by AA (Angle - Angle) similarity criterion.

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. In \( \triangle MKL \), side \( MK = 16 \), \( ML = 11 \). In \( \triangle PNO \), side \( PN = 46 \), \( PO = x \). The corresponding sides are \( MK \) and \( PN \), \( ML \) and \( PO \). So, the proportion is \( \frac{ML}{PO}=\frac{MK}{PN} \), which is \( \frac{11}{x}=\frac{16}{46} \).

Step3: Solve for \( x \)

Cross - multiply: \( 16x = 11\times46 \). Calculate \( 11\times46 = 506 \). Then, \( x=\frac{506}{16}=31.625 \approx 31.6 \) (rounded to the nearest tenth).

Answer:

\( x = 31.6 \)