 # 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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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?

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.

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

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.

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

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

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.

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

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

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.

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

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

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

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.

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.

Example Does D lie on the perpendicular bisector of

Example

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

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

WARM-UP

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.

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.

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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