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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.

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carlosego
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)

Answer:

The coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].

Step-by-step 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

[tex]Centroid=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})[/tex]

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

[tex]Centroid=(\frac{-4-2+5}{3},\frac{1-3-2}{3})[/tex]

[tex]Centroid=(\frac{-1}{3},\frac{-4}{3})[/tex]

Therefore the coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].

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