1-3 Vocabulary coordinate plane midpoint segment bisector leg hypotenuse midpoint formula distance formula

A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) . The location, or coordinates, of a point are given by an ordered pair (x, y).

The midpoint M of AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3. New Definitions: midpoint → 2  OR 2  → midpoint bisects → 2  OR 2  → bisects

Example 5: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. E D 4x + 6 7x – 9 F S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R S T –2x –3x – 2

Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7).

Check It Out! Example 2 S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T.

You can also use the Pythagorean Theorem to find the distance between two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c.

PROOF WILL BE IN CHAPTER 5

Use the Pythagorean Theorem to Derive the Distance Formula: B(x2, y2) A(x1, y1)

The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane. d =   2

Example 3: Using the Distance Formula Find FG and JK. Then determine whether FG  JK.

Example 4: Finding Distances in the Coordinate Plane Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(–2, –5).

Check It Out! Example 4b Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R to S. R(–4, 5) and S(2, –1)

Example 5: Sports Application A player throws the ball from first base to a point located between third base and home plate and 10 feet from third base. What is the distance of the throw, to the nearest tenth?

Lesson Quiz: Part I 1. Find the coordinates of the midpoint of MN with endpoints M(-2, 6) and N(8, 0). 2. K is the midpoint of HL. H has coordinates (1, –7), and K has coordinates (9, 3). Find the coordinates of L. 3. Find the distance, to the nearest tenth, between S(6, 5) and T(–3, –4). 4. The coordinates of the vertices of ∆ABC are A(2, 5), B(6, –1), and C(–4, –2). Find the perimeter of ∆ABC, to the nearest tenth.

Lesson Quiz: Part II 5. Find the lengths of AB and CD and determine whether they are congruent.