Answer in Geometry for Mira #100158

Question #100158

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

Expert's answer

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: $2\sqrt{5}.$

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Comments

Assignment Expert

03.07.20, 21:13

Dear April, thank you for describing new methods of solving and comments.

April

03.07.20, 04:39

There are ways to do this problem without solving x and y. There's a theorem for this exact scenario that would go BC^2+AC^2=5*AB^2. Or you could use the median length formula, in which you would need substitution with but don't need to fully solve x and y.