1.3 Distance and Midpoints1.3 Distance and Midpoints

Objectives:Objectives:

Find the distance between two points Find the distance between two points using the distance formula and using the distance formula and Pythagorean’s Theorem.Pythagorean’s Theorem.

Find the midpoint of a segment.Find the midpoint of a segment.

Distance Between Two Distance Between Two PointsPoints In the last section we learned that In the last section we learned that

whenever you connect two points you whenever you connect two points you create a segment.create a segment.

We also learned every segment has a We also learned every segment has a distance.distance.

The The distancedistance between two pointsbetween two points,, or the distance of a segment, is or the distance of a segment, is determined by the number of units determined by the number of units between the two points. between the two points.

Distance Formula on a Number Distance Formula on a Number LineLine

If a If a segment is on a number line, we simply segment is on a number line, we simply find its length by using the find its length by using the Distance Distance FormulaFormula which states the distance which states the distance between two points is the between two points is the absolute value absolute value of the differenceof the difference of the values of the two of the values of the two points.points.

| A – B | = | B – A | = Distance | A – B | = | B – A | = Distance

Use the number line to find QR.

The coordinates of Q and R are –6 and –3.

Answer: 3

Distance Formula

Simplify.

Example 1:Example 1:

Use the number line to find AX.

Answer: 8

Your Turn:Your Turn:

Distance Formula on a Coordinate Distance Formula on a Coordinate PlanePlane

Segments may also be drawn on Segments may also be drawn on coordinate planes. To find the coordinate planes. To find the distance distance between two points on a coordinate between two points on a coordinate planeplane with coordinates with coordinates (x(x11, y, y11)) and and (x(x22, , yy22)) we can use this formula: we can use this formula:

Distance Formula on a Coordinate Distance Formula on a Coordinate PlanePlane

… … or we can use the Pythagorean or we can use the Pythagorean Theorem.Theorem.

The The Pythagorean TheoremPythagorean Theorem simply simply states that the square of the hypotenuse states that the square of the hypotenuse equals the sum of the squares of the two equals the sum of the squares of the two legs.legs.

aa22 + b + b2 2 = c= c22

Find the distance between E(–4, 1) and F(3, –1).

Pythagorean Theorem Method

Use the gridlines to form a triangle so you can use the Pythagorean Theorem.

Example 2:Example 2:

Pythagorean Theorem

Simplify.

Take the square root of each side.

Example 2:Example 2:

Distance Formula Method

Distance Formula

Simplify.

Simplify.

Answer: The distance from E to F is units. You can use a calculator to find that is approximately 7.28.

Example 2:Example 2:

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

Answer:

Your Turn:Your Turn:

Midpoint of a SegmentMidpoint of a Segment

The The midpointmidpoint of a segment is the of a segment is the point halfway between the endpoints point halfway between the endpoints of the segment. If X is the midpoint of the segment. If X is the midpoint of AB, then AX = XB.of AB, then AX = XB.

To find the midpoint of a segment on To find the midpoint of a segment on a number line find ½ of the sum of a number line find ½ of the sum of the coordinates of the two endpoints.the coordinates of the two endpoints.

a + ba + b22

The coordinates of J and K are –12 and 16.

Answer: 2

Simplify.

The coordinates on a number line of J and K are –12 and 16, respectively. Find the coordinate of the midpoint of .

Let M be the midpoint of .

Example 3:Example 3:

Midpoint of a SegmentMidpoint of a Segment

If the segment is on a coordinate If the segment is on a coordinate plane, we must use the plane, we must use the midpoint midpoint formula for coordinate planesformula for coordinate planes which states givenwhich states given a segment with a segment with endpoints endpoints (x(x11, y, y11)) and and(x(x22, y, y22)) the midpoint is… the midpoint is…

M= ( xM= ( x11 + x + x22 , y , y11 + y + y22 ) ) 2 22 2

Let G be and H be .

Answer: (–3, 3)

Find the coordinates of M, the midpoint of ,

for G(8, –6) and H(–14, 12).

Example 4:Example 4:

a. The coordinates on a number line of Y and O are 7 and –15, respectively. Find the coordinate of the midpoint of .

b. Find the coordinates of the midpoint of for X(–2, 3) and Y(–8, –9).

Answer: (–5, –3)

Answer: –4

Your Turn:Your Turn:

More About MidpointsMore About Midpoints

You can also find the coordinates of You can also find the coordinates of an endpoint of a segment if you an endpoint of a segment if you know the coordinates of the other know the coordinates of the other endpoint and its midpoint. endpoint and its midpoint.

Let F be in the Midpoint Formula.

Find the coordinates of D if E(–6, 4) is the midpoint of and F has coordinates (–5, –3).

Write two equations to find the coordinates of D.

Example 5:Example 5:

Solve each equation.

Answer: The coordinates of D are (–7, 11).

Multiply each side by 2.

Add 5 to each side.

Multiply each side by 2.

Add 3 to each side.

Example 5:Example 5:

Answer: (17, –11)

Find the coordinates of R if N(8, –3) is the midpoint of and S has coordinates (–1, 5).

Your Turn:Your Turn:

Multiple-Choice Test ItemWhat is the measure of if Q is the midpoint of ?

A B 4 C D 9

Example 6:Example 6:

Read the Test Item

Solve the Test Item

Because Q is the midpoint, you know that .

Use this equation and the algebraic measures to find a

value for x.

You know that Q is the midpoint of , and the figure gives algebraic measures for and . You are asked to find the measure of .

Example 6:Example 6:

Definition of midpoint

Distributive Property

Subtract 1 from each side.

Add 3x to each side.

Divide each side by 10.

Example 6:Example 6:

Answer: D

Original measure

Simplify.

Now substitute for x in the expression for PR.

Example 6:Example 6:

Answer: B

Multiple-Choice Test ItemWhat is the measure of if B is the midpoint of ?

A 1 B 3 C 5 D 10

Your Turn:Your Turn:

Assignment:Assignment:

Geometry:Geometry:

Pg. 25 – 26, #13 – 28, 31 – 40, 43 Pg. 25 – 26, #13 – 28, 31 – 40, 43 - 44- 44

Pre-AP Geometry: Pre-AP Geometry: Pg. 25 – 26, #13 - 45Pg. 25 – 26, #13 - 45