Question

Complete the following proof.

Prove: In an equilateral triangle the three medians are equal.

Answer 1

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Answer 2

Explanation:

In the figure attached,

ΔABC is an equilateral triangle,

Sides AB = BC = AC and points P, Q, and R are the midpoints of these sides respectively.

If the coordinates of A(0, 0), B(2a, 0) and C(a, b)

AB = 2a

AC =
`√(a^2+b^2)`

Since AB = AC

2a =
`√(a^2+b^2)`

4a² = a² + b²

3a² = b²

Therefore, ordinate pairs representing midpoints of AB, BC and AC will be

P =
`((2a+0)/(2),(0)/(2))` =(a, 0)

Q =
`((a+2a)/(2),(b)/(2))` =
`((3a)/(2),(b)/(2))`

R =
`((a+0)/(2),(b+0)/(2))` =
`((a)/(2),(b)/(2))`

Now we will find the lengths of medians with the help of formula of distance between two points (x, y) and (x', y')

d =
`√((x-x')^2+(y-y')^2)`

AQ =
`\sqrt{(0-(b)/(2))^2+(0-(3a)/(2))^2}`

=
`\sqrt{(b^2)/(4)+(9a^2)/(4)}`

=
`(1)/(2)(√(b^2+9a^2))`

=
`(1)/(2)√(12a^2)` [Since b² = 3a²]

=
`a√(3)`

BR =
`\sqrt{(2a-(a)/(2))^(2)+(0-(b)/(2))^2}`

=
`\sqrt{((3a)/(2))^2+(-(b)/(2))^2}`

=
`(1)/(2)√(b^2+9a^2)`

=
`(1)/(2)(√(12a^2))`

=
`a√(3)`

CP = b =
`a√(3)`

Therefore, AQ = BR = CP =
`a√(3)`

Hence, medians of an equilateral triangle are equal.