Key concepts Line continue infinitely in both
directions, their length cannot be measured.
A Line Segment is a part of line that is noted by two end points (x1, y1) and (x2, y2).
The length of a lie segment can be found using the distance formula.
Midpoint The midpoint of a line segment is the
point on the segment that divides it into two equal parts.
Find the midpoint of a line segment is like finding the average of the two endpoints.
Midpoint formula The midpoint formula is used to find the
midpoint of a line segment. The midpoint formula is
Proving Midpoints You can prove that the midpoint is
halfway between the endpoints by calculating the distance from each endpoint to the midpoint.
EXAMPLE Calculate the midpoint of the line
segment with endpoints of (-2,1) and (4,10).
First determine the endpoints of the line segment (in this case the points given)
Second, substitute the values of (x1, y1) and (x2, y2) into the midpoint formula
Prove mathematically: Calculate the distance between the
endpoint (-2, 1) and the midpoint (1, 5.5)
Use the distance formula
Step two Calculate the distance between the
other endpoint and the midpoint.
If the distance is the same () Then you have proven that (1, 5.5) is
the midpoint of the line segment.
Finding other points. Determine the point that is ¼ the
distance from the endpoint (-3, 7) of the segment with the endpoints of (-3, 7) and (5, -9)
Step two Calculate the difference between the x-
values. distance between x values substitute the x values simplify 8
Step four The x value is to the right of the original
endpoint, therefore add the product to the x-value of the endpoint.
This is the x-value of the point with the given ratio.
(-3) + 2 = -1
Step five Calculate the difference between the y
values. distance between y values substitute the x values simplify 16
The y value is down from the original endpoint, therefore subtract the product from the y-value of the endpoint.
7-4 = 3 The point that is ¼ the distance from
the endpoint (-3,7) of the segment (-3,7) and (5,-9) is (-1,3)
Now you try: Determine the point that is 2/3 the distance from the endpoint (2,9) Of the segment with endpoints (2,9) and (-4,-6)
Find an endpoint A line segment has one endpoint at
(12,0) and a midpoint (10, -2). Locate the second endpoint.
Step one Substitute the values of (x1, y1) into the
midpoint formula and simplify, midpoint formula Substitute (12,0)
Find the value of X The midpoint (10, -2) is equal to
Set up an equation to find the value of x = 10 equation Now solve for x
Find the value of y Create an equation to find the value of
y. = -2 equation Y+0 = -4 Multiply both sides by 2 Y = -4 Simplify.
Calculate area of a triangle 1. find the equation of the line that
represents the base of the triangle. 2. Find the equation of the line that
represents the height of the triangle. 3.Find the point of intersection of the
line representing the height and the line representing the base.
4. Calculate the length of the base of the triangle (distance formula).
5. Calculate the height of the triangle(distance formula).o 6. Calculate the area using the formula:o A = ½ bh
Guided example triangle withvertices A(1, -1) B(4,3) C(5, -3) Let AC be the base. Slope for this line is: M=(-3)-(-1) = -2 = -1 (5)-(1) 4 2
Write the equation for AC y – y1 = m(x-x1) point slope form
Substitute -1/2 for m, and (1, -1) for (x1, y1)
Y –(-1) = -1/2(X – 1) Simplify Y + 1 = -1/2x + ½ Isolate y: y = -1/2x -1/2
Equation for height This equation needs a slope
perpendicular to the base:
Slope will be 2 Use point slope form and point (4,3) to
write the equation.