Question

The right triangle is given. Let the length of the hypotenuse be 10, and the lengths of
the legs a and b. If the length of the median from the vertex of right angle to the
opposite side is Vab, the length of the height from the same vertex is

A. 2
B. 2.5
C. 3
D. 3.5
​

Answer 1

By midline theorem,

$a^{2} +b^{2} =50+2\sqrt{ab}^{2}$

$100=50+2ab$

$2ab=50$

$ab=25$

Let the height be h

therefore,

$\frac{1}{2}ab=\frac{1}{2}$×10×h

$\frac{1}{2}ab=5h$

$h=\frac{ab}{10}$

$h=\frac{25}{10}$

$h=2.5$

Therefore the correct answer is 2.5(option B).

Answer 2

$\sqrt{ab \: } = \frac{10}{2} = 5$

The length of the height = √ab/10

$\frac{25}{10} = 2.5$

or

By midline theoram

a² + b² = 50 + 2√ ab²

100 = 50 + 2ab

2ab. = 50

ab = 25

let height be L

$\frac{1}{2} ab = \frac{1}{2} \times 10 \times l$

$\frac{1}{2} ab \: = \frac{5}{l }$

$l \: = \frac{ab}{10}$

$l = \frac{ 25}{10}$

$l \: = 2.5$

therefore B i.e 2.5 is correct

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