Question

Prove that the median to the hypotenuse of a right triangle is half the hypotenuse. Proof: By the Distance Formula , MN=___ Therefore OP = 1/2MN

Answer 1

MN

`= √(4a^2+4b^2)= 2√(a^2+b^2)`

Explanation:

From the figure attached,

ΔMON is a right triangle and coordinates of the points M and N are M(0, 2b) and N(2a, 0).

Coordinates of midpoint P →
`((2a+0)/(2), (0+2b)/(2))`

From the formula of the distance between two points,

`d = \sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2}`

`\texttt {MN} = √((2a-0)^2+(0-2b)^2)\\\\ = √(4a^2+4b^2)\\\\ = 2√(a^2+b^2)`

Similarly, OP

`= √((0-a)^2+(0-b)^2)\\\\ = √(a^2+b^2)`

Therefore, OP

`= (1)/(2)(MN)`

and MN

`= √(4a^2+4b^2)= 2√(a^2+b^2)`