contestada

In rectangle abcd, points p and q lie on side AB and DC respectively. Angle PMQ is a right angle, M is the midpoint of side BC and PB=4/3 BC. What is the ration PM: MQ

In rectangle abcd points p and q lie on side AB and DC respectively Angle PMQ is a right angle M is the midpoint of side BC and PB43 BC What is the ration PM M class=

Respuesta :

jacob193

Answer:

PM:MQ = 8:3.

Step-by-step explanation:

[tex]\rm \angle B\hat{M}P + 90^{\circ} + \angle C\hat{M}Q = 180^{\circ}[/tex];

[tex]\implies \rm \angle B\hat{M}P + \angle C\hat{M}Q = 90^{\circ}[/tex];

[tex]\implies \rm 90^{\circ} - \angle B\hat{M}P = \angle C\hat{M}Q[/tex].

Also,

[tex]\rm \angle B\hat{P}M = 90^{\circ} - \angle B\hat{M}P[/tex] in right triangle PBM.

Thus [tex]\rm \angle{P\hat{B}M} = \angle C\hat{M}Q[/tex].

Additionally [tex]\rm \angle \hat{B} = 90^{\circ} = \angle \hat{C}[/tex].

Therefore [tex]\rm \triangle PBM \sim \triangle MCQ[/tex].

[tex]\rm \displaystyle BC = 2\;MC[/tex] for M is the midpoint of segment BC.

[tex]\rm \displaystyle PB = \frac{4}{3}BC = \frac{8}{3}MC[/tex].

[tex]\rm \triangle PBM \sim \triangle MCQ[/tex] implies that

[tex]\displaystyle \rm PM:MQ = PB:MC = 1:\frac{8}{3} = 8:3[/tex].

Otras preguntas