Question

In a triangle ABC is a right triangle right angled at C. A line through the mid-point
Mof hypotenuse AB and parallel to BC intersects AC at D. Show that
MD is perpendicular to AC​

Answer 1

In △ABC,

(1) M is the mid point of AB and

MD∥BC

D is the mid point of AC

(2) As MD∥BC &

AC is transversal

∠MDC+∠BCD=180°

∠MDC+90 ∘ =180 ∘

⇒∠MDC=90 ∘

⇒MD⊥AC

(3) In △AMD and △CMD

AD=CD

∠ADM=∠CDM

DM=DM

△AMD≅△CMD

AM=CM⇒AM= 1/2AB

⇒CM=AM= 1/2 AB

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