Section 1.9 Distance and Midpoint Formulas and Circles

The Distance Formula

Distance Formula Cont. The distance, d, between the points (x1, y1) and (x2, y2) in the rectangular coordinate system is To compute the distance between the two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root is the distance between the two points.

Find the distance between (4,-5) and (9,-2). Example 1 Find the distance between (4,-5) and (9,-2). Page 250 Problems 1-18

The Midpoint Formula Consider a line segment whose endpoints are (x1, y1) and (x2, y2). The coordinates of the segment’s midpoints are To find the midpoint, take the average of the two x- coordinates and the the average of the two y-coordinates. Find the midpoint of the segment whose endpoints are (-1,5) and (6,8). (-1,5) (6,8)

Example 2 Find the midpoint of the segment whose endpoints are (-1,7) and (-5,9). Page 250 Problems 19-30

Repetitive Practice will help you memorize the formulas! So… open your books to page 250 and complete problems 1 – 29 odds.

Circles!!! We would use distance formula to find the radius of this circle using the center point and any point from the edge. We would use midpoint formula to find the center point of this circle using any two points from the edge that are opposite of each other.

Circles A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from the circle’s center to any point on the circle is called the radius. The Standard Form of the Equation of a Circle With center of (h, k) and radius r is…

Write the standard form of the equation of the circle with center (-4,1) and radius of 3. y Standard Form 9 8 7 6 5 4 Domain: [-7, -1]; 3 3 2 Range:[-2, 4] (-4,1) 1 x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

Find the center and radius of the circle whose equation is Graph the equation. Use the graph to identify the relation’s domain and range. Why is it a relation and not a function? -1 1 2 3 4 5 6 7 8 9 10 -8 -7 -6 -5 -4 -3 -2 x y Center (-3, 4) Radius = 3 Domain: [-6,0]; 3 Range:[1,7] (-3,4) -7 -6 -5 -4 -3 -2

Page 250 Prob 31 – 45 odds Example 3 Write the standard form of the equation of the circle with center at (-2,7) and a radius of 5. Page 250 Prob 31 – 45 odds Page 250 Problems 31-40

Center: (6, -5) Radius 7 Example 4 Find the center and radius of the circle whose equation is below. Graph the equation. Use the graph to identify the relation’s domain and range. Center: (6, -5) Radius 7 Domain: [-1, 8]; Range:[-12, 2] Page 250 Problems 41-52

Please turn to Page 250 Complete problems 2, 6, 18, 22, 24, 30, 36, 38, 44, 48

Graphing Calculator Notes

General Form

Center: Radius (2, -6) 5 Example 5 Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation. Center: (2, -6) Radius 5

Center: Radius (-3, 4) 5 Example 6 Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation. Center: (-3, 4) Radius 5 Page 250 Problems 53-64

Review Time Add these problems to your notes paper to help you review! Additional Practice problems can be found on page 250-251 problems 65-82

(d) 1. Find the distance between the points (-1,8) and (9,5). (a) (b)

2. Find the midpoint of the line segment with the endpoints of (-1,-1) and (- 5,8). (b) (c) (d) (a)

3. Write the standard form of a circle with center at (0,7) and a radius of 4. (b) (c) (d) (c)