In the triangle ABC AC = 3, BC = 4, the medians AK and VL are perpendicular to each other.

| June 24, 2021 | education

In the triangle ABC AC = 3, BC = 4, the medians AK and VL are perpendicular to each other. Find the area of the triangle ABC.

Property of the medians: if the medians are perpendicular, then the sum of the squares of the sides on which they are dropped is five times the third side.

In our triangle, the median AK goes down to the BC side, and the OHL median goes down to the AC side.

It follows from the property:

BC ^ 2 + AC ^ 2 = 5 * AB ^ 2.

4 ^ 2 + 3 ^ 2 = 5 * AB ^ 2

AB = √ ((16 + 9) / 5) = √ (25/5) = √5.

To find the area, Heron’s formula is used:

S = √ (p * (p – a) * (p – b) * (p – c)), where p is a semiperimeter.

p = (3 + 4 + √5) / 2 = (7 + √5) / 2

S = √ ((7 + √5) / 2 * ((7 + √5) / 2 – 3) * ((7 + √5) / 2 – 4) * ((7 + √5) / 2 – √ 5)) = √ (7 + √5) / 2 * (7 + √5 – 6) / 2 * (7 + √5 – 8) / 2 / (7 + √5 – 2 * √5) / 2) = √ ((7 + √5) / 2 * (1 + √5) / 2 * (√5 – 1) / 2 * (7 – √5) / 2) = √ (((7 + √5) * (7 – √5)) / 4 * ((√5 – 1) * (√5 + 1)) / 4) = √ ((49 – 5) / 4 * (5 – 1) / 4) = √ ( 44/4 * 1) = √11.



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