Distance & Midpoint Pythagorean Theorem: square of the hypotenuse.
Notes # Alpha 4
In a right triangle, the sum of the squares of each leg equals the
c
a
a2 + b2 = c2
b
Distance Formula for Two Points: The distance, d units, between two points with coordinates (x1, y1) and (x2, y2) is given by the following formula. d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
Slope: the slope of a line is the ratio of the change in the y-values of the coordinates of the points to the corresponding change in x-values. The slope of a line is constant. The slope, m, of the line through (x1, y1) and (x2, y2) is given by the following equation, y − y1 m= 2 if x1 ≠ x2. x2 − x1 Midpoint of a Line Segment: If the coordinates of A and B are (x1, y1) and (x2, y2), respectively, then the midpoint of AB has coordinates x + x 2 y1 + y 2 M = 1 , 2 2 Ex A: Find the distance between the points with the given coordinates. Then, find the slope of the line passing through each pair of points. Finally, find the midpoint of the segment with endpoints the given coordinates.
#1) d=
(4, -3), (-1, -7) m=
M=
Note: You must write down the formula every time you use it. Don’t skip ANY steps.
Linear Relationship & Functions Page 1 of 2
Distance & Midpoint
Ex A: Find the distance between the points with the given coordinates. Then, find the slope of the line passing through each pair of points. Finally, find the midpoint of the segment with endpoints the given coordinates. #2)
(b, 9 + a), (b, a + 6)
d=
m=
M=
Ex B: Determine whether the figure with vertices with given coordinates is a parallelogram.
#1)
(2, 3), (5, 1), (7, 6), (4, 8)
#2)
(5, 12), (9, 15), (5, 20), (1, 16)
Ex C: Collinear points lie on the same line. Find the value of k for which points with each set of coordinates is collinear.
#1)
(4, 0), (k, 3), (4, -3)
Analytic Geometry:
#2)
(5, -2), (-2, 5), (1, k)
Note: First find the slope of the two points with numbers. Then use the slope formula with the point with k in it and any other point.
The study of coordinate geometry from an algebraic perspective. The formulas for slope, midpoint, and distance are used in analytic geometry. Linear Relationship & Functions Page 2 of 2