In the coordinate plane

MATHEMATICS · Multi-part · CBSE · Mathematics · Coordinate Geometry · Coordinate Geometry

Problem

In the coordinate plane, let

A(0,0)

and

B(b,0)

with

b>0

be fixed points on the

x

–axis. Two points

C(x_1,y_1)

and

D(x_2,y_2)

with

y_1>0

and

y_2>0

lie above the

x

–axis, forming triangles

\triangle ABC

and

\triangle ABD

on the common base

AB

. (i) Prove that

\displaystyle \text{area}(\triangle ABC):\text{area}(\triangle ABD)=y_1:y_2

. (ii) Explain geometrically why this ratio depends only on the

y

–coordinates of

C

and

D

and not on their

x

–coordinates, provided both

x_1

and

x_2

lie between

0

and

b