6.1 Perpendicular and Angle Bisectors

Geogebra Warm-up

With geogebra, try to create a diagram of two triangles sharing a side where the longest side out of the five sides is not opposite the largest angle.

Open a new Geogebra File1) Construct a line segment AB. 2) Construct a perpendicular bisector (4th from left,

3rd down) through that segment. 3) Place a point C on the perpendicular bisector. 4) Construct segments AC and BC. 5) Two finger click on AC, click on object properties,

click show label: value. Repeat for BC. 6) Move point C with the arrow tool and make a

conjecture about a point on a perpendicular bisector.

Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Proof

The converse is also true.

Perpendicular Bisector Converse

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Perpendicular Bisector of a Triangle

A line that divides one side of a triangle into two congruent parts and is perpendicular to that

side

If there’s a point on the Perpendicular Bisector, then it is equidistant to the endpoints of the segment.

Open a new Geogebra File

1) an angle (8th from left, top choice). 2) rays (3rd from left, 4th down) as the sides of the

angles. 3) an angle bisector(4th from left, 4th down)4) two perpendicular lines (4th from left, top choice),

each through a side of the angle and through the SAME POINT on the angle bisector.

5) construct a circle with a center at the “SAME POINT” in the previous step through a point where a perpendicular line intersects with a side of the angle.

Construct:

Angle Bisector of a TriangleA segment that lies on an angle bisector and

that has one endpoint at the vertex and the other on the opposite side of the triangle

If there’s a point on the Angle Bisector, then it is equidistant from the sides of the angle.

The converse is also true.

Altitude of a TriangleA perpendicular segment from a vertex to the

base or to the line containing the base

Median of a TriangleA line segment connecting a vertex of a triangle

to the midpoint of the opposite side

Midsegment of a TriangleA line segment connecting the midpoints of two

sides of the triangle

Properties of the Midsegment of a Triangle

1 – The midsegment is half the length of the base

2 – The base and the midsegment are parallel.e parallel

Sketch each of the following in a triangle with the proper markings

• Perpendicular Bisector• Angle Bisector• Median• Altitude• Midsegment

Write the equation of the perpendicular bisector given the endpoints of the segment it bisects.

1) (-2, 3) and (4, 1)2) (-1, -5) and (3, -1)3) (12, -5) and (18, 5)