Question

Crater Lake Oregon is a roughly circular lake. The lake basin formed about 7000 years ago when the top of a volcano exploded in an immense explosion. Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2). Find the coordinates of the center of the lake.

Answer 1

The coordinates of the center of the lake are
`((-1)/(3),(-4)/(3))`.

Explanation:

It is given that Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2).

If we joint these points, then we get a triangle and the center of a triangle is known as centroid.

The formula for centroid of a triangle is

`Centroid=((x_1+x_2+x_3)/(3),(y_1+y_2+y_3)/(3))`

The centroid of the triangle H(-4,1), G(-2,-3), and C(5,-2) is

`Centroid=((-4-2+5)/(3),(1-3-2)/(3))`

`Centroid=((-1)/(3),(-4)/(3))`

Therefore the coordinates of the center of the lake are
`((-1)/(3),(-4)/(3))`.

Answer 2

To solve the problem we must find the equation of the circumference of the form:
(Xa) ² + (Yb) ² = r²
They do not give us the center or the radio, but we have 3 points.
With the points H (-4,1), G (-2, -3) and C (5, -2) we find 3 equations
1) (-4-a) ² + (1-b) ² = r²
2) (-2-a) ² + (-3-b) ² = r ²
3) (5-a) ² + (-2-b) ² = r ²
Now we have a system of 3 equations with 3 unknowns: r, a, b
When solving the system we have to:
r = 5
a = 1
b = 1
Therefore, the center of the circumference of the lake is the point (a, b)
That is, the point (1,1)