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Using properties of Midsegments Suppose you are given only the three midpoints of the sides of a triangle. Is it possible to draw the original triangle?

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2 Guided Practice: The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle. 1. Plot the midpoints on a coordinate plane.

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3 Guided Practice: continued 2. Connect the midpoints to form the midsegments,, and.

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4 Guided Practice: continued 3. Calculate the slope of each midsegment. Calculate the slope of. The slope of is Slope formula Substitute (–2, 5) and (3, 1) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

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5 Guided Practice: continued Calculate the slope of. The slope of is 7. Slope formula Substitute (3, 1) and (4, 8) Simplify.

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6 Guided Practice: continued Calculate the slope of. The slope of is Slope formula Substitute (–2, 5) and (4, 8) Simplify.

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7 Guided Practice: 4. Draw the lines that contain the midpoints. The endpoints of each midsegment are the midpoints of the larger triangle. Each midsegment is also parallel to the opposite side.

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8 Guided Practice: continued The slope of is From point Y, draw a line that has a slope of

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9 Guided Practice: continued The slope of is 7 From point X, draw a line that has a slope of 7

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10 Guided Practice: continued The slope of is From point Z, draw a line that has a slope of The intersections of the lines form the vertices of the triangle.

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Properties of Triangles Perpendicular and Angle Bisectors Objective: To use properties of perpendicular bisectors and angle bisectors

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Perpendicular Bisector Perpendicular Bisector – a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

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Equidistant Equidistant from two points means that the distance from each point is the same.

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Perpendicular Bisector Theorem Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

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Converse of the Perpendicular Bisector Theorem Converse of the Perpendicular Bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.

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Example Does D lie on the perpendicular bisector of

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Example

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Distance from a point to a line The shortest distance from one point to another is a straight line.

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Examples Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?

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WARM-UP

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Angle Bisector Theorem Angle Bisector Theorem – If a point (D) is on the bisector of an angle, then it is equidistant from the two sides of the angle.

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Converse of the Angle Bisector Theorem Converse of the Angle Bisector Theorem – If a point is on the interior of an angle, and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

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Examples Does the information given in the diagram allow you to conclude that P is on the angle bisector of angle A?

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