Question

In the pulley system shown in this figure, MQ = 30 mm, NP = 10 mm, and QP = 21 mm. Find MN.

Answer 1

MN is 63 mm.

Since triangles MPQ and NQP are similar, we have the following

proportion:

`(MQ)/(NP) = (QP)/(MN)`

Substituting the given values, we have:

`(30)/(10) = (21)/(MN)`

Solving for MN, we get:

`MN = (21 * 30)/(10) = 63 mm`

Therefore, MN is 63 mm.

Answer 2

`\boxed{\overline{MN}=37.96}`

Explanation:

For a better understanding of this problem, see the figure below. Our goal is to find
`\overline{MN}`. Since:

`\angle MRS=\angle MQP=90^(\circ) \\ \\ \overline{MQ}=\overline{MR}=30mm`

and
`\overline{MN}` is a common side both for ΔMRN and ΔMQN, then by SAS postulate, these two triangles are congruent and:

`\overline{RN}=\overline{QN}`

By Pythagorean theorem, for triangle NQP:

`\overline{QN}=\sqrt{\overline{NP}^2+\overline{QP}^2} \\ \\ \overline{QN}=√(10^2+21^2) \\ \\ \overline{QN}=√(541)`

Applying Pythagorean theorem again, but for triangle MQN:

`\overline{MN}=\sqrt{\overline{MQ}^2+\overline{QN}^2} \\ \\ \overline{MN}=\sqrt{30^2+(√(541))^2} \\ \\ \boxed{\overline{MN}=37.96}`