Asked by Samantha
The coordinates of the endpoints of the hypotenuse of a right triangle are (7, 5) and (3, 1). Find the other vertex. There are two possible solutions.
Answers
Answered by
Reiny
actually, there is an infinite number of solutions
one of the properties of a circle is that a triangle is formed with the diameter as a base and the other point on the circle, you will always have a right angle.
so the centre would be (5,3) and the radius would be 2√2
equation:
(x-5)^2 + (y-3)^2 = 8
so now pick any x
e.g.
x = 4
1 + (y-3)^2 = 8
(y-3)^2 = 7
y = 3 ± √7 ----> 2 points , (4, 3+√7) and (4, 3-√7)
x = 5
0 + (y-3)^2 = 8
y-3 ± 2√2
y = 3 ± 2√2 ---> 2 more points (5,3+2√2) and (5, 3-2√2)
....
x = 6.5
y -3 = ±√5.75
y = 3 ± √5.75)
---- > 2 more points (6.5 , 3 + √5.75) and (6.5, 3-√5.75)
So far i have 6 such points,
btw, use your calculator to check that the slope of the 2 sides are negative recriprocals of each other, (or their product is -1)
It works for all of them, I did it.
one of the properties of a circle is that a triangle is formed with the diameter as a base and the other point on the circle, you will always have a right angle.
so the centre would be (5,3) and the radius would be 2√2
equation:
(x-5)^2 + (y-3)^2 = 8
so now pick any x
e.g.
x = 4
1 + (y-3)^2 = 8
(y-3)^2 = 7
y = 3 ± √7 ----> 2 points , (4, 3+√7) and (4, 3-√7)
x = 5
0 + (y-3)^2 = 8
y-3 ± 2√2
y = 3 ± 2√2 ---> 2 more points (5,3+2√2) and (5, 3-2√2)
....
x = 6.5
y -3 = ±√5.75
y = 3 ± √5.75)
---- > 2 more points (6.5 , 3 + √5.75) and (6.5, 3-√5.75)
So far i have 6 such points,
btw, use your calculator to check that the slope of the 2 sides are negative recriprocals of each other, (or their product is -1)
It works for all of them, I did it.