Distance and Midpoints

You are vacationing in an unfamiliar place. You ask the clerk in the hotel lobby where the nearest coffee shop is, within walking distance.

one

block?

2 or 3

blocks? An hour on foot?

Is “walking distance” a universally understood term? Is it an accurate

measure?

What is walking distance?

A mile or two?

Essential Questions

When does distance make a difference in life?

Can distance ever be negative?

Coordinate Plane

Graph the following points:point A (1,3) point B (-2,4)

point C (-4, -3) point D (2,-5)

Distance on a number line

What is PQ if a=1, b=5?What if a=-10, b=3?

Distance Formula

2 22 1 2 1( ) ( )d x x y y

The distance between any two points with coordinates and

is given by the formula1 1( , )x y 2 2( , )x y

Essential Questions

Can distance ever be negative?

Why or why not?

Example 1 Find the distance from A(-4,-3) to B(2, 5) using the Distance Formula.

Pythagorean Theorem

In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

SmallLeg2 + LargeLeg22 = Hypontenuse2

Example 2Find the distance from H(2,3) to K(-3,-1) by using the Pythagorean Theorem.

Just beyond Midway, PA

Midpoint FormulaMidpoint—the point halfway

between the endpoints of a segment

Examples• The coordinates on a number line of J and

K are -12 and 16 find the midpoint of line segment JK

• Point G has the coordinate (8, -6) , the midpoint has the coordinate (-3, 3) find the coordinate of the end of the line segment.