Question

1. prove that the median to the hypotenuse of a right triangle is half the hypotenuse. plan: since mid - points will be involved, use multiples of _ to name the coordinates for m and n. given: △mno is a right triangle with right ∠mon. p is the midpoint of mn. prove: op = 1/2mn

Explanation:

Step1: Assign coordinates

Let \(M=(0,2a)\) and \(N=(2b,0)\) since mid - points are involved and using multiples of 2 simplifies calculations.

Step2: Find the mid - point \(P\) of \(MN\)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(P = (\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For \(M=(0,2a)\) and \(N=(2b,0)\), we have \(P=(\frac{0 + 2b}{2},\frac{2a+0}{2})=(b,a)\).

Step3: Calculate the distance \(MN\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(M=(0,2a)\) and \(N=(2b,0)\), \(MN=\sqrt{(2b - 0)^2+(0 - 2a)^2}=\sqrt{4b^{2}+4a^{2}} = 2\sqrt{a^{2}+b^{2}}\).

Step4: Calculate the distance \(OP\)

For \(O=(0,0)\) and \(P=(b,a)\), using the distance formula \(OP=\sqrt{(b - 0)^2+(a - 0)^2}=\sqrt{a^{2}+b^{2}}\).

Step5: Compare \(OP\) and \(MN\)

We can see that \(OP=\frac{1}{2}MN\).

Answer:

The median \(OP\) to the hypotenuse \(MN\) of right - triangle \(MNO\) is half the hypotenuse \(MN\).