6 1 Perpendicular and Angle Bisectors Geogebra Warmup

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 File 1) Construct a line segment AB. 2) Construct a perpendicular bisector (4 th from left, 3 rd 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 Construct: 1) an angle (8 th from left, top choice). 2) rays (3 rd from left, 4 th down) as the sides of the angles. 3) an angle bisector(4 th from left, 4 th down) 4) two perpendicular lines (4 th 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.

Angle Bisector of a Triangle A 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 Triangle A perpendicular segment from a vertex to the base or to the line containing the base

Median of a Triangle A line segment connecting a vertex of a triangle to the midpoint of the opposite side

Midsegment of a Triangle A 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

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)