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Midpoints of line segments

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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.

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Distance Formula

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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.

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Midpoint formula

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Proving Midpoints You can prove that the midpoint is halfway between the endpoints by calculating the distance from each endpoint to the midpoint.

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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

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Example cont.

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Prove mathematically:

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Step two

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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)

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Step one Draw the segment on a coordinate plane.

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Step two

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Step three Multiply the difference by the given ratio of ¼ (8)(1/4) = 2

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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

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Step five

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Multiply the difference by the given ratio (1/4) (16)(1/4) = 4

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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)

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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)

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Find an endpoint A line segment has one endpoint at (12,0) and a midpoint (10, -2). Locate the second endpoint.

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

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Step one

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Find the value of X

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Find the value of y

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

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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.

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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

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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

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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

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Equation for Base AC Y = - ½ x – ½

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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.

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Equation is Y=2x - 5

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Find the point of intersection Set the two equations equal to each other and solve for x

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

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

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Find length of AC Use the distance formula

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Length of AC

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

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Calculate the area

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Area of triangle ABC Is 11 units

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