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.
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.
To find the other vertex of the right triangle, we need to determine the length and slope of the hypotenuse. Then, using the length and slope, we can calculate the coordinates of the other vertex.
Step 1: Calculate the length of the hypotenuse.
To find the length of the hypotenuse, we can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (7, 5) and (3, 1), we substitute x1 = 7, x2 = 3, y1 = 5, and y2 = 1 into the formula:
Distance = √((3 - 7)^2 + (1 - 5)^2)
= √((-4)^2 + (-4)^2)
= √(16 + 16)
= √32
= 4√2
So, the length of the hypotenuse is 4√2.
Step 2: Calculate the slope of the hypotenuse.
The slope of a line can be found using the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Using the coordinates (7, 5) and (3, 1), we substitute x1 = 7, x2 = 3, y1 = 5, and y2 = 1 into the formula:
Slope = (1 - 5) / (3 - 7)
= (-4) / (-4)
= 1
So, the slope of the hypotenuse is 1.
Step 3: Calculate the coordinates of the other vertex.
We now have the length of the hypotenuse (4√2) and the slope (1). We'll use these values to calculate the coordinates of the other vertex.
Let the coordinates of the other vertex be (x, y).
Since the hypotenuse is the longest side of a right triangle, the other two sides should be shorter. Therefore, the y-coordinate of the vertex should be either greater than 5 or less than 1.
Let's assume the y-coordinate is greater than 5.
Using the slope-intercept form of a line (y = mx + b), we can write the equation of the hypotenuse as:
y = 1x + b
Now, substituting the coordinates (7, 5) into the equation:
5 = 1(7) + b
b = 5 - 7
b = -2
So, the equation of the hypotenuse is y = x - 2.
Now, we'll use the equation of the hypotenuse and the length of the hypotenuse (4√2) to find the x-coordinate of the other vertex.
Substituting the y-coordinate (greater than 5) into the equation:
y = 4√2 - 2
Now, we have both the x and y coordinates of the other vertex.
Therefore, one possible solution is (x, y) = (4√2, 4√2 - 2).
Similarly, you can calculate the other possible solution by subtracting 4√2 from the y-coordinate to get a y-coordinate less than 1.
Therefore, the other possible solution is (x, y) = (4√2, -2 - 4√2).
So, there are two possible solutions for the other vertex of the right triangle.