Question

foot perpendicular theorm​

Answer 1

Foot of perpendicular :

Theorem: If PQ is perpendicular to a plane XY and if from Q, the foot of the perpendicular, a straight line QR is drawn perpendicular to any straight line ST in the plane, then PR is also perpendicular to ST.

converse of the theorem

Construction: Through Q draw in the plane XY the straight line LM parallel to ST.

Proof: Since LM is parallel to ST and QR perpendicular to ST hence, QR is perpendicular to LM. Again, PQ is perpendicular to the plane XY; hence, it is perpendicular to the line LM. Therefore, LM is perpendicular to both PQ and QR at Q. This implies LM is perpendicular to the plane PQR. Now, ST and LM are parallel and LM is perpendicular to the plane PQR; hence, ST is perpendicular to the plane PQR. Therefore, ST is perpendicular to PR or in other words, PR is perpendicular to ST.

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