Question

In an equilateral triangle, show that three times the square of a side equals four times the square of medians.

Answer 1

Step-by-step explanation:

Given:

△ABC is an Equilateral Triangle.

AB=BC=AC

CD Perpendicular to AB

To Prove:

3AC

2

=4CD

2

Proof:

In △ADC

By Pythagoras Theorem,

AC

2

=AD

2

+DC

2

.....(1)

In △BDC

By Pythagoras Theorem,

BC

2

=DC

2

+BD

2

.....(2)

Adding 1) and 2) We get,

2AC

2

=2DC

2

+AD

2

+BD

2

(SinceAC=BC)

Since, AD=BD=

2

1

AC (SinceAC=AB)

∴ 2AC

2

=2DC

2

+

2

1

AC

2

∴ 2AC

2

−

2

1

AC

2

=2DC

2

∴ 3AC

2

=4CD

2

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