# Find the height of a right-angled triangle from the vertex of the right angle if it divides

Find the height of a right-angled triangle from the vertex of the right angle if it divides the hypotenuse into 4cm and 16cm lengths.

In a right-angled triangle AB and AC – legs, BC – hypotenuse, AH – height.

BH = 4 cm; CH = 16 cm;

By the Pythagorean theorem

from tr – ka ABC: BC² = AB² + AC²;

from tr – ka AVN: AB² = BH² + AH²;

from tr – ka АСН: АС² = АН² + CH²;

Of the two last. express .:

AB² + BC² = BH² + AH² + AH² + HC²; AB² + BC² = 2 * AH² + BH² + CH²;

2 * AH² = (AB² + BC²) – BH² – CH²;

AB² + BC² = (BH + CH) ² = (4 + 16) ² = 400 (cm²);

BH² = 4² = 16 (cm²); CH² = 16² = 256 (cm²);

2 * AH² = 400 – 16 – 256 = 128 (cm); AH² = 64 cm²;

AH = √64 = 8 (cm).

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