Question

In Trapezium PQRS; Side PQ || Side SR; AR = 5AP; AS = 5AQ then prove that; SR = 5PQ.

Answer 1

A trapezium PQRS is drawn in such a way that PQ || RS , AR = 5AP and AS = 5AQ
See In figure, ∵ PQ || RS
∴ ∠PQA = ∠ASR
∠QPA = ∠ARS
And also ∠PAQ = ∠SAR

From A - A - A similarity rule,
∆APQ ~ ∆ARS
∴ PQ/RS = AP/AR = AQ/AS
Given, AR = 5AP so,
PQ/RS = AP/5AP = 1/5
⇒PQ = RS/5
⇒RS = 5PQ or, SR = 5PQ
Hence, proved$\bf{SR=5PQ}$

Answer 2

PROOF:-

IN TRAPEZIUM PQRS,

SIDE PQ II SIDE SR____(GIVEN)

ANGLE PQA = ANGLE ASR___(VERTICALLY OPPOSITE)

Angle QPA =Angle ARS__(ALTERNATE INTERIOR ANGLE

ANGLE PAQ=ANGLE SAP__(ALTERNATE INTERIOR ANGLE)

∆APQ~ ∆ARS__(BY SSS TEST OF SIMILARITY)

AP upon AR= PQ upon SR= AQ upon AS

APupon 5AP=PQ upon SR__(AP=5AP)

1 upon 5=PQ upon SR

SR=5PQ_____(HENCE PROVED)