Question

Find the midpoint of the hypotenuse.

Answer 1

12
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Answer 2

The length of the hypotenuse PQ is n√2, and the midpoint is (n/2, n/2).

To find the length of the hypotenuse PQ in the right-angled triangle POQ, we can use the Pythagorean Theorem. The theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, PO and OQ are the legs of the right triangle, and PQ is the hypotenuse.

The coordinates of P, O, and Q are (0, n), (0, 0), and (n, 0) respectively. The lengths of PO and OQ are the differences in their y-coordinates and x-coordinates, respectively.

PO = |0 - n| = n

OQ = |n - 0| = n

Now, applying the Pythagorean Theorem:

PQ^2 = PO^2 + OQ^2

PQ^2 = n^2 + n^2

PQ^2 = 2n^2

`\[ PQ = √(2n^2) = n√(2) \]`

So, the length of the hypotenuse PQ is n√2.

To find the midpoint of the hypotenuse, we take the average of the coordinates of P and Q:

Midpoint (xm, ym) = ((xP + xQ)/2, (yP + yQ)/2)

Midpoint = (n/2, n/2)