The Distance Formula & Equations of Circles Proving the distance formula Proving equation of circles Examples of finding the distance between two points Examples of equations of circles

Proving the Distance Formula If you have two points and are trying to find the distance between those points, you use the Pythagorean Theorem. So, using the Pythagorean Theorem (a2 + b2 = c2), you can assume that α2 + β2 = µ2 So, the distance from P to Q is written: d(P,Q) = (x1,y1) (x2,y2) (x2,y1) α β Q P µ Where α = x2-x1 andβ = y2-y1 and µ = distance from point P to point Q

Examples Find the distance between the points (2,4) & (3,-5) sqrt[(3-2)2 + (-5-4)2] = sqrt(1 + 81) = sqrt(82) (2,4) (3,-5)

Examples cont. Find the distance between the points (-5,-5) & (4,-3) sqrt[(-5-4)2 + (-5-(-3))2] = sqrt(81 + 4) = sqrt(85) (4,-3) (-5,-5)

Examples cont. Find the distance between the points (1,4) & (7,4) sqrt[(1-7)2 + (4-4)-2] = sqrt(36 + 0) = sqrt(36) = 6 (7,4) (1,4)

Equation of Circles With a circle with a radius r, centered at point P, all points that are r units away from P make the circle. If the coordinates of P are (h,k), you can use the distance formula to get If we square this, we get the standard equation of a circle: (x-h)2 + (y-k)2 = r r P r (h,k)

Examples Draw the graph of the equation (x-2)2 + (y+2)2 = 9 (h,k) = (2,-2) r = 3

Examples cont. Draw the graph of the equation (x+3)2 + (y+4)2 = 4 (h,k) = (-3,-4) r = 2