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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.