Homework Lesson 9.1 page 567 #22-27 ALL Lesson 1-3: Formulas 1

Lesson 9.1 distance and midpoint formula Lesson 1-3: Formulas2

3 The Distance Formula The distance d between any two points with coordinates and is given by the formula d =.

Example 1: 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).

Method 1 Use the Pythagorean Theorem. Count the units for sides a and b. Example 1 Continued a = 5 and b = 9. c 2 = a 2 + b 2 = = = 106 c = 10.3

Example 1 Continued Method 2 Use the Distance Formula. Substitute the values for the coordinates of D and E into the Distance Formula.

Example 2: Using the Distance Formula Find the length for FG and JK. Then determine whether FG  JK. Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)

Example 2 Continued Step 2 Use the Distance Formula.

Check It Out! Example 3 Find EF and GH. Then determine if EF  GH. Step 1 Find the coordinates of each point. E(–2, 1), F(–5, 5), G(–1, –2), H(3, 1)

Check It Out! Example 3 Continued Step 2 Use the Distance Formula.

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? Example 4: Sports Application

Set up the field on a coordinate plane so that home plate H is at the origin, first base F has coordinates (90, 0), second base S has coordinates (90, 90), and third base T has coordinates (0, 90). The target point P of the throw has coordinates (0, 80). The distance of the throw is FP. Example 5 Continued

Lesson 1-3: Formulas 13 Midpoint Formula M = Find the midpoint between (-2, 5) and (6, 4) x 1 = -2, x 2 = 6, y 1 = 5, and y 2 = 4 Example: In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are.

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). = (–5, 5)

Check It Out! Example 1 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).

Example 2: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y). Step 2 Use the Midpoint Formula:

Example 2 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by = 2 + x Simplify. – 2 10 = x Subtract. Simplify. 2 = 7 + y – 7 –5 = y The coordinates of Y are (10, –5).

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. Step 1 Let the coordinates of T equal (x, y). Step 2 Use the Midpoint Formula:

Check It Out! Example 2 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. –2 = –6 + x Simplify = x Add. Simplify. 2 = –1 + y = y The coordinates of T are (4, 3).

Circle Lesson 1-3: Formulas 20

Find the center of the circle Given a diameter with endpoints A(0,3) and B(5,1) Lesson 1-3: Formulas 21

Find the area of the circle Given a diameter with endpoints A(0,3) and B(5,1) Lesson 1-3: Formulas 22

Find the circumference of the circle Given a diameter with endpoints A(0,3) and B(5,1) Lesson 1-3: Formulas 23

Find the center, area, and circumference of the circle Given a diameter with endpoints F(-4,5) and E(-2,1) Lesson 1-3: Formulas 24

Lesson 1-3: Formulas 25 Slope Formula Find the slope between (-2, -1) and (4, 5).Example: Definition:In a coordinate plane, the slope of a line is the ratio of its vertical rise over its horizontal run. Formula:The slope m of a line containing two points with coordinates and is given by the formula where.

Lesson 1-3: Formulas 26 Describing Lines Lines that have a positive slope rise from left to right. Lines that have a negative slope fall from left to right. Lines that have no slope (the slope is undefined) are vertical. Lines that have a slope equal to zero are horizontal.

Lesson 1-3: Formulas 27 Some More Examples Find the slope between (4, -5) and (3, -5) and describe it. Since the slope is zero, the line must be horizontal. m = Find the slope between (3,4) and (3,-2) and describe the line. m = Since the slope is undefined, the line must be vertical.

Lesson 1-3: Formulas 28 Example 3 : Find the slope of the line through the given points and describe the line. (7, 6) and (– 4, 6) Solution: This line is horizontal. x y (7, 6) up 0 left 11 (-11) (– 4, 6) m

Lesson 1-3: Formulas 29 Example 4: Find the slope of the line through the given points and describe the line. (– 3, – 2) and (– 3, 8) Solution: This line is vertical. x y (– 3, – 2) up 10 right 0 (– 3, 8) undefined m

Lesson Quiz: Part I (17, 13) (3, 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 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.

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