Question

prove that perpendicular drawn from the vertex to the opposite sides are concurrent​

Answer 1

Given: ΔABC in which AL, BM and CN are the perpendicular drawn on the sides BC, CA and AB respectively.

To Prove: AL, BM and CN are concurrent.

Construction: Through A, B, and C, draw lines QR, RP and PQ parallel to BC, CA and AB respectively.

Proof:

In quadrilateral ABCQ,

AQ || BC  (Construction)

QC || AB  (Construction)

∴ ABCQ is a parallelogram      (A quadrilateral is a parallelogram, if both pair of its opposite sides are parallel)

⇒ AQ = BC    ...(1)    (Opposite sides of parallelogram are equal)

Similarly, ARBC is a parallelogram.

∴ AR = BC     ...(2)    (Opposite sides of parallelogram are equal)

Form (1) and (2), we get

AQ = AR       ...(3)

AL ⊥ BC and RQ || BC   ...(4)

AL is the perpendicular bisector of RQ.

Similarly, BM and CN are perpendicular bisector of sides PR and PQ respectively.

∴ AL, BM and CN are perpendicular bisectors of the sides of ΔPQR which are concurrent.

Hence, AL, BM and CN are concurrent.

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