Midpoints of line segments

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.

Distance Formula

√ (𝑥2−𝑥1 )2+(𝑦 2−𝑦 1)2 . .

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

Example cont.

Substitute numbers in:

Simplify: = (1, 5.5)

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 one Draw the segment on a coordinate

plane.

Step two Calculate the difference between the x-

values. distance between x values substitute the x values simplify 8

Step three Multiply the difference by the given ratio

of ¼ (8)(1/4) = 2

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

Multiply the difference by the given ratio (1/4)

(16)(1/4) = 4

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.

Analyze problem One endpoint is (12,0) Midpoint is (10,-2) The other endpoint is unknown

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

= 10

x + 12 = 20 Multiply both sides by 2 X = 8 Subtract 12 from both

sides.

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.

The endpoint of the segment with one endpoint at (12,0) and a midpoint at (10, -2) is (8, -4)

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 Base AC Y = - ½ x – ½

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.

Equation is

Y=2x - 5

Find the point of intersection Set the two equations equal to each

other and solve for x

Substitute value of x in to find y Substitute 9/5 into either equation

Point of intersection is (9/5. -7/5)

Find length of AC Use the distance formula

Length of AC Is 25 units

Find length of height From point B to the intersection

(4,3) (9/5, -7/5)

Height is 11 5

Calculate the area

Area of triangle ABC Is 11 units