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A right triangle is shown in the graph.

Part A: Use the Pythagorean Theorem to derive the standard equation of the circle, with center at (f, g) and a point on the circle at (x, y). Show all necessary math work.

Part B: If (f, g) = (3, –1) and h = 8, determine the domain and range of the circle.

Part C: Is the point (10, –4) inside the border of the circle if (f, g) = (3, –1) and h = 8? Explain using mathematical evidence.

A right triangle is shown in the graph Part A Use the Pythagorean Theorem to derive the standard equation of the circle with center at f g and a point on the ci class=

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sqdancefan

Answer:

  A:  (x -f)^2 +(y -g)^2 = h^2

  B:  domain: [-5, 11]; range: [-9, 7]

  C:  yes, inside

Step-by-step explanation:

Part A:

Use of the Pythagorean theorem gets you to the equation for a circle in essentially one step:

  sum of squares of sides = square of hypotenuse

  (x -f)^2 +(y -g)^2 = h^2 . . . . . . circle centered on (f, g) with radius h

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Part B:

The circle will be defined for values of x in the domain f ± h, and for values of y in the range g ± h.

  domain: 3 ± 8 = [-5, 11]

  range: -1 ±8 = [-9, 7]

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Part C:

The distance from point (10, -4) to (f, g) is ...

  h^2 = (10 -3)^2 +(-4 -(-1))^2

  h^2 = 7^2 +(-3)^2 = 49 +9 = 58

  h = √58 < 8 . . . . the distance to the point is less than h=8.

The point is inside the circle.