Question

PQ has a length of 17 units with P(-4,7). If the x- and y-coordinates of Q are both greater than the x- and y-coordinates of P, what are possible integer value coordinates of Q? Explain.

Let PQ be the hypotenuse of a right triangle that also has a horizontal leg and a vertical leg. The hypotenuse then has length 17, and a Pythagorean triple can be used to say the shorter leg has length
and the longer leg has length. If the shorter leg is horizontal, then Q is described by the ordered pair. If the shorter leg is vertical, then Q is described by the ordered pair

Answer 1

Explanation:

a distance on a coordinate grid is always the Hypotenuse of a right-angled triangle with the differences of the x coordinates and the y coordinates as the legs.

so, per Pythagoras

distance² = (x diff)² + (y diff)²

we have here a distance 17. 17² = 289.

to get integer coordinates the lengths of the legs must be integer numbers.

what 2 squared integer numbers can be added to get 289 ?

I found after subtracting 4, 9, 16, 25, 36, 49 and finally 64 (the squares of 2, 3, 4, 5, 6, 7, 8) that

289 - 64 = 225, which is also a square number (15²).

so, we get the possible integer coordinates of Q to be

(-4 + 8, 7 + 15) = (4, 22)

in this case the shorter leg has the length 8, and the longer leg the length 15.

the shorter leg is in this case horizontal, because it is the x coordinate difference. and Q is again as described (4, 22).

if we make the shorter leg vertical, that means it is the y coordinate difference. and Q is then

(-4 + 15, 7 + 8) = (11, 15)