The legs of a right triangle are 3: 4. Find the line segments into which the hypotenuse

| January 13, 2021 | education

The legs of a right triangle are 3: 4. Find the line segments into which the hypotenuse of this triangle is divided by the height drawn from the right angle, if the length of the hypotenuse is 10 cm.

1. Vertices of triangle A, B, C. AB – hypotenuse. BC and AC – legs. CH – height.
2. By the condition of the problem BC / AC = 3/4. BC = 3AC / 4.
3. BC² + AC² = AB² (by the Pythagorean theorem).
4. Substitute in this equation ВС = 3АС / 4:
(3АС / 4) ² + АС² = 100.
25АС² = 1600.
AC² = 64.
AC = 8 cm.
BC = 3 x 8/4 = 6 cm.
5. We calculate the area (S) of the triangle ABC:
S = BC x AC / 2 = 6 x 8/2 = 24 cm².
6. Calculate the length of the CH height using another formula for calculating the area of ​​a triangle:
S = AB x CH / 2.
CH = 2S / AB = 2 x 24/10 = 4.8 cm.
7. Calculate the length of the segment AH using the Pythagorean theorem:
AH = √AC² – CH² = √64 – 23.04 = 6.4 cm.
8. We calculate the length of the HV segment:
BH = AB – AH = 10 – 6.4 = 3.6 cm.
Answer: AH = 6.4 cm, BH = 3.6 cm.



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