1-5: Midpoints and Segment Congruence

1-5: Midpoints and Segment Congruence Expectations: G1.1.5: Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint. G1.1.6: Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, and plane), axioms, definitions, and theorems. 1.5: Midpoints

Midpoint • Defn: M is the midpoint of AB iff M is ____________ A and B and __________. x x B A M 1.5: Midpoints

Locating the midpoint • a. Locate A and B anywhere on your paper. • b. Draw AB. • c. Fold A onto B and crease your paper. • d. The intersection of your crease and is the midpoint of AB. 1.5: Midpoints

Midpoint on a Number Line • a. Draw a number line with at least 10 integers on it. • b. Locate A and B on your number line. • c. Draw AB. • d. Fold your paper to find the midpoint of AB. • e. Develop an algebraic method for finding the midpoint of a segment on a number line. 1.5: Midpoints

Midpoint on a Coordinate Grid • a. Locate A and B on a coordinate grid s.t. all coordinates are even. • b. Draw AB. • c. Fold your paper to determine the midpoint. • d. Develop an algebraic method to determine the coordinates of the midpoint of AB. 1.5: Midpoints

Midpoint Formula • Part 1: On a number line, the coordinate of the midpoint of a segment whose endpoints are a and b is ______ . • Ex: What is the coordinate of the midpoint of a segment on a number line with endpoints –4 and 12? 1.5: Midpoints

Midpoint Formula • Part 2: On a coordinate grid, the coordinates of the midpoint of a segment whose endpoints are (x1, y1) and (x2, y2) are 1.5: Midpoints

What is the midpoint of the line segment between points (2,6) and (3,8)? • (2, 2.5) • (2.5, 2) • (5, 4.5) • (4, 5.5) • (2.5, 7) 1.5: Midpoints

If M is the midpoint of AB, A(4,10) and M(-2,6), what are the coordinates of B? 1.5: Midpoints

Segment Bisector • Defn: A line or a segment is a segment bisector iff it contains the _________ of the segment. l Y A M B Z YZ and l are bisectors of AB. 1.5: Midpoints

Congruent Segments • Defn: Segments with endpoints A and B and C and D are congruent, written AB ≅ CD, iff 1.5: Midpoints

Congruent Segments • If AB ≅ CD, A B C D These marks are used to show the segments are congruent. 1.5: Midpoints

Congruent Segments • If AB ≅ CD, 10 cm ? cm A B C D then AB = CD. 1.5: Midpoints

Congruent Segments • If AB = CD, 14 cm 14 cm A B C D 1.5: Midpoints

Congruent Segments • If AB = CD, A B C D then 1.5: Midpoints

The Midpoint Theorem • If M is the midpoint of AB, then AM ≅ MB. • How is this different than the definition of a midpoint? 1.5: Midpoints

In the figure below, P is the midpoint of AB, AP = 5x – 8 and PB = 16 – 3x. Determine the value of x and the length of AB. X A P Y B 1.5: Midpoints

A proof is a logical argument that is justified by one or more mathematical statements known to be true. Two types of proof are two column proof and paragraph proof. 1.5: Midpoints

Prove the Midpoint Theorem • Given: M is the midpoint of AB. • (We assume the given to be true.) • Prove: AM ≅ MB 1.5: Midpoints

1.5: Midpoints

If B is the midpoint of AC and D is the midpoint of BC, prove AB = 2CD. A B D C 1.5: Midpoints

1.5: Midpoints

Assignment • pages 41-43, • # 19, 21, 27, 29, 33 and 47 1.5: Midpoints

1-5: Midpoints and Segment Congruence