The median and height of a right-angled triangle drawn to the hypotenuse are 50 cm

The median and height of a right-angled triangle drawn to the hypotenuse are 50 cm and 48 cm, respectively. Find the sides of the triangle.

Let us denote the triangle given by the condition ABC, angle C – straight line, CH – height, CM – median.
In a right-angled triangle, the median is half the hypotenuse, in our case we get:
CM = AB / 2 → AB = 2 * CM = 100 (cm).
In a right-angled triangle CHM, we find the leg HM (according to the Pythagorean theorem):
HM = √ (CM² – CH²) = √ (2500 – 2304) = √196 = 14 (cm).
AH = AM + NM = 50 + 14 = 64 (cm).
In a right-angled triangle AHC we find the hypotenuse AС:
AC = √ (AH² + CH²) = √ (4096 + 2304) = √6400 = 80 (cm)
In the triangle ABC we find the leg BC:
BC = √ (AB² – AC²) = √ (10000 – 6400) = √3600 = 60 (cm).
Answer: 60 cm, 80 cm, 100 cm – sides of the ABC triangle.

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