Geometry

13.1 The Distance Formula

Example 1 Find the distance between the two points.a. A(–5, 4) and B(–1, 4) b. C(2, –5) and D(2, 7)

A(-5, 4). . B(-1, 4)

4 units

As you can see, if two points share an x coordinate or a y coordinate, the distance between the two points can be found by finding the distance (the absolute value of the difference)

between the other coordinates.

| -5 – (-1) | | 7 – (-5) |

12 units

What if two points do not lie on a horizontal or vertical line?

When two points do not lie on a horizontal or vertical line, you can find the distance between points by using the distance formula which is derived from the Pythagorean Theorem.

1 1( , )x y 2 2( , )x y

2 22 1 2 1( ) ( )d x x y y

The distance d between points and is:

Example 2Find the distance between (–3, 4) and (1, –4).

Why? Let’s try an example to find out!

22 )4(413

64168054

(-3, 4).

. (1, -4)

4

8

Pythagorean Theorem!

4√5

Use the distance formula to find the distance between the two points.4. (4, 2) and (1,-1)

Answer: 3√2

Example 3Given points A(1, 1), B(–3, –11), and C(4, 10), find AB, AC, and BC. Are A, B, and C collinear? If so, which point lies between the other two?

AB = √(1+3)2 + (1+11)2

= √16 + 144

= √160

= 4√10

BC = √(-3 – 4)2 + (-11 - 10)2

= √49 + 441

= √490

= 7√10

AC = √(1 – 4)2 + (1 – 10)2

= √9 + 81

= √90

= 3√10

Are A, B, and C collinear? . . .If yes, the two small distances add to the large distance.

4√10 + 3√10 = 7√10

The points are collinear!!Since BC is the longest distance, B and C are the endpoints.

Thus, point A must be in the middle.

Pick one, #1 or #2!!

12. Show that the triangle with vertices P(5, 0), E(–1, –2), and T(3, 6) is isosceles.

PE = √(5+1)2 + (0+2)2

= √36 + 4

= √40

= 2√10

ET = √(-1 – 3)2 + (-2 – 6)2

= √16 + 64

= √80

= 4√5

PT = √(5 – 3)2 + (0 – 6)2

= √4 + 36

= √40

= 2√10

The triangle is isosceles because two of its sides are the same length!

13. Quadrilateral GEMA has vertices G(3, 8), E(8, –3), M(–2, –5), and A(–5, 2). Show that its diagonals are congruent.

GM = √(3+2)2 + (8+5)2

= √25 + 169

= √194

No need to simpilfy…

EA = √(8+5)2 + (-3 – 2)2

= √169 + 25

= √194

The diagonals are congruent!

2 2 2( ) ( )x a y b r An equation of the circle with center (a, b) and radius r is:

Why is the circle formula in the section about the distance formula?

If you rearrange the formula above it reads…

22 )()( byaxr Now that we know the distance formula, this represents all points that are “r distance” away from a point (a, b).

This is precisely a circle!!!

This is the circle formula to remember!!

2 2 2( ) ( )x a y b r An equation of the circle with center (a, b) and radius r is:

Let’s analyze (x – 0)2 + (y – 0)2 = 81 to see if it really is a circle!!

How could this be a circle?

3Write an equation of the circle that has the given center and radius.14. C(0, 0); r = 9 15. C(-3, -8); r = 6 16. C(1, -2); r =

Example 4Find the center and radius of the circle with the equation:

2 2( 3) ( 5) 4x y

Center: (-3, 5) Radius = 2

x2 + y2 = 81 (x + 3)2 + (y + 8)2 = 36 (x – 1)2 + (y + 2)2 = 3

2 2( 2) ( 4) 9x y 2 2( 7) ( 3) 25x y

2 2( 1) 49x y 2 2 9( 1) ( 2)

16x y

Find the center and radius of each circle. Sketch the graph.

17. 18.

19. 20.

Center: (2, -4)Radius = 3

.

Center: (-7, -3)Radius = 5

.

Center: (0, 1)

Radius = 7

. Center: (1, 2)

Radius = 3/4 .

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