Answer to Question #100158 in Geometry for Mira

Question

In triangle ABC, median AD is perpendicular to median BE. Find AB if BC=6 and AC=8

Answer 1

Let O be the intersection point of the medians. Medians are divided in the ratio 2: 1 at the intersection, according to the property of medians.

Let Y = OD, X = OE.

Then OB = 2X, OA = 2Y.

Compose the system:

Y² + 4X² = 9

X² + 4Y² = 16

Express X = $\sqrt{16-4Y^2}$ ;

Then

Y² = 9-4 * (16-4Y²) = 9-64 + 16Y²

15Y² = 55

$Y=\sqrt{\frac{11}{3}}, \,$ $X=\sqrt{16-4 \cdot \frac{11}{3}}=\sqrt{\frac{4}{3}}$

For side AB we get the equation:

AB² = 4 (X² + Y²) = 4 * (4/3 + 11/3) = 4 * 15/3 = 20

$AB=\sqrt{20}=2\sqrt{5}$ . Answer: $2\sqrt{5}.$