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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 (x 1, y 1 ) and (x 2, y 2 ).  The length of a lie segment can be found using the distance formula.

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

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 (x 1, y 1 ) and (x 2, y 2 ) into the midpoint formula

Example cont.

Prove mathematically:

Step two

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

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

 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

Find the value of X

Find the value of y

 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 with vertices 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 – y 1 = m(x-x 1 ) point slope form  Substitute -1/2 for m, and (1, -1) for (x 1, y 1 )  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

Find length of height  From point B to the intersection  (4,3) (9/5, -7/5)

 Calculate the area

Area of triangle ABC  Is 11 units

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