Derivation & Application of Distance & Midpoint Formulas in Coordinate Geometry, Slides of Algebra

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§10.1 Distance

MIdPoint Eqns

Review §

 Any QUESTIONS About

• §9.6 → Exponential Decay & Growth

 Any QUESTIONS About HomeWork

• §9.6 → HW-

9.6 MTH 55

Pythagorean Distance

• Now consider any two points ( x 1 , y 1 ) and ( x 2 , y 2 ).
• These points, along with ( x 2 , y 1 ), describe a right triangle. The lengths of the legs are | x 2 – x 1 | and | y 2 – y 1 |.

Pythagorean Distance

• Find d , the length of the hypotenuse, by using the Pythagorean theorem: d^2 = | x 2 – x 1 |^2 + | y 2 – y 1 | 2
• Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses: d^2 = ( x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2

Example  Find Distance

• Find the distance between (3, 1) and (5, −6). Find an exact answer and an approximation to three decimal places.
• Solution: Substitute into the distance formula

d = (5 − 3) 2 + −( 6 −1)^2

= (2) 2 + −( 7)^2 = 53 ≈ 7.280.

Substituting

This is exact. Approximation

Example  Verify Rt TriAngle

• Let A (4, 3), B (1, 4) and C (−2, −4) be three points in the plane. Connect these Dots to form a Triangle, Then: a. Sketch the triangle ABC b. Find the length of each side of the triangle c. Show that ABC is a right triangle.

Example  Verify Rt TriAngle

• Soln b. Find the length of each side of the triangle → Use Distance Formula d A ( , B ) = (^) ( 4 − (^1) )^2 + (^) ( 3 − (^4) )^2 = 9 + 1 = 10

d B ( , C ) = (^) ( 1 − −( 2 )) 2

• (^) ( 4 − −( 5 )) 2

= 9 + 81 = 90

d B ( , C ) = (^)  4 − −( 2 )

2

• (^)  3 − −( 5 )

2

= 36 + 64 = 100 = 10

Example  Verify Rt TriAngle

• Soln c.: Show that ABC is a Rt triangle.
• Check that a^2 + b^2 = c^2 holds in this triangle, where a , b , and c denote the lengths of its sides. The longest side, AC , has length 10 units.

^ d A ( ,^ B )

2

• (^)  d B ( , C )

2 = 10 + 90

= 100 = (^) ( (^10) )^2 = (^)  d A ( , C )

2 .  It follows from the converse of the Pythagorean Theorem that the triangle ABC IS a right triangle.

Example  BaseBall Distance

• Solution: conveniently choose home plate as the origin and place the x -axis along the line from home plate to first base and the y -axis along the line from home plate to third base

Example  BaseBall Distance

• Find from the DiagramThe coordinates of home plate ( O ), first base ( A ) second base ( C ) and third base ( B )

The MidPoint Formula

• Now that we have derived the Distance

formula from the Pythagorean Theorem

we use the distance formula to develop a

formula for the coordinates of the

MidPoint of a segment connecting two

points.

The MidPoint Formula

• If the endpoints of a segment are ( x 1 , y 1 ) and ( x 2 , y 2 ), then the coordinates of the midpoint are

(^1 2) , 1 2. 2 2

^ x^ +^ x^ y^ + y    ( x 1 , y 1 )

( x 2 , y 2 )

x

y

 That is, to locate the midpoint, average the x -coordinates and average the y -coordinates

CIRCLE Defined

• A circle is a set of points in a Cartesian

coordinate plane that are at a fixed

distance r from a specified point ( h , k ).

• The fixed distance r is called the radius of

the circle, and

• The specified point ( h , k ) is called the

center of the circle.

CIRCLE Graphed

• The graph of a circle with center ( h , k ) and radius r.