Question

50 Points! Multiple choice geometry question. Photo attached. Thank you!

Answer 1

b)
`MN = 6(3)/(4)`

Explanation:

In ΔPQN and ΔLMN,

∠PQN = ∠LMN (given)

∠PNQ = ∠LNM (vertically opposite angles)

Since two angles in the triangles are same, the third angle must alos be same

Therefore,

∠QPN = ∠MLN

By AAA property, the two triangles are similar

Therefore,

`(PQ)/(LM) =(QN)/(MN) =(PN)/(LN) \\\\\implies (8)/(6) =(9)/(MN) =(PN)/(LN) \\\\\implies (8)/(6) =(9)/(MN)\\\\\implies MN =(9*6)/(8)\\\\\implies MN =(54)/(8)\\\\\implies MN =(27)/(4)\\\\\implies MN =6(3)/(4)`