10.5 Distance Formula

Pythagorean Theorem In a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. (“c” is always the hypotenuse) a b c (leg) (hyp)

Distance Formula Where d stands for distance x1 & y1 are one endpoint of a segment x2 & y2 are the second endpoint of a segment (x1 , y1) (x2 , y2) d

What do you say when you walk into a cold room? It’s collinear! (It’s cold in here!)

Midpoint of a Segment The point halfway between the endpoints of a segment If X is the midpoint of Segment AB , then (the measure of AX = the measure of XB) A B X

Midpoint Formula On a Double Number Line (Coordinate Plane): On a Single Number Line: x y (5,5) (-3,-1) 5 Units (x1,y1) (x2,y2) A B -3 7 X 2 (1,2)

(a single number line) M

(a double number line) (Not to scale) M(2.5, 1.5)

Find the Coordinates of an Endpoint The Midpoint Formula can be used to find the coordinates of an endpoint when the midpoint and one endpoint are given. Find the coordinates of Endpoint B, if M = 5 is the midpoint, and A = -12 is the other endpoint. 22 17 units (a single number line) A -12 M 5 B ? Multiply both sides by 2 Solve for B

(The midpoint is always the ordered pair with no subscripts) Let X = (x1,y1) (One endpoint) Y = (-2, 2) (Midpoint) Z = (2, 8) (Other endpoint) (x2, y2) (x, y) (x, y) (x2, y2) (x1, y1) (The midpoint is always the ordered pair with no subscripts) X(?) Y(-2,2) Z(2, 8) (-4, -6)

-2x Does it work? +5 Look for an equation to write, then solve: Be sure to answer the question.

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