Midpoints, bisectors and trisectors

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Presentation transcript:

Midpoints, bisectors and trisectors I CAN… Define and identify midpoint, angle bisector and trisector Write proofs involving bisectors and trisectors

Draw a figure in which… A, B, and C are collinear A, D, and E are collinear B, C, and D are noncollinear F is between A and E F is between R and S A, E, R and S are noncollinear

Draw a figure in which… A, K, O and Y are collinear K is between O and A The length of AO added to the length of AY is equal to the length of OY (OA + AY = OY) A is to the right of O

Midpoint: a point that divides a segment into two congruent segments. The midpoint, M, of is the point between A and B such that AM = MB. Alternate Definition: The point on a line segment that is equidistant from its endpoints. NOTE: rays and lines cannot have midpoints. Why?

Bisect: cut into two equal parts. We can also say that a midpoint bisects a line segment... Bisect: cut into two equal parts. Segment Bisector: a segment, ray, line or plane that intersects a segment at its midpoint. bisects C G L P

ANGLE BISECTOR - For ray QR to be the angle bisector of PQS, point R must be on the interior of PQS and PQR must be congruent to RQS. In other words, a ray must divide an angle into two congruent, adjacent angles.

Trisect: cut into three equal parts. Segments and angles can also be trisected... Trisect: cut into three equal parts. Trisection points: the two points where a segment is divided. Angle trisectos: the two rays that divide an angle A R S C If what conclusions can we draw? G M GEO is trisected by the two interior rays. What conclusions can we draw? E R O

JK bisects HJL. Given that mHJL = 42°, what are the measures of HJK and KJL?

In the diagram, MO bisects LMN In the diagram, MO bisects LMN. The measures of the two congruent angles are (3x – 20)° and (x + 10)°. Solve for x and find the measure of all three angles.

You are given that T and W are trisection points on SP and SP = 24. Find ST. Find SW. S T W P

Given OM = 2x + 3, MP = x – 9 and OP = 45. Is M the midpoint of OP?

Prove: H is the midpoint of DF G F Given: Prove: H is the midpoint of DF H D E STATEMENTS REASONS