Apply the formula for midpoint. Use the distance formula to find the distance between two points.
description
Transcript of Apply the formula for midpoint. Use the distance formula to find the distance between two points.
![Page 1: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/1.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Apply the formula for midpoint.Use the distance formula to find the distancebetween two points.
Objective
![Page 2: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/2.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
midpoint
Vocabulary
![Page 3: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/3.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
In Lesson 5-4, you used the coordinates of points to determine the slope of lines. You can also use coordinates to determine the midpoint of a line segment on the coordinate plane.
The midpoint of a line segment is the point that divides the segment into two congruent segments. Congruent segments are segments that have the same length.
You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.
![Page 4: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/4.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
![Page 5: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/5.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2).
Substitute.
Write the formula.
Simplify.
Additional Example 1: Finding the Coordinates of a Midpoint
G(–4, 3)
H(6, -2)
![Page 6: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/6.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 1
Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).
Write the formula.
Substitute.
Simplify.
E(–2, 3)
F(5, –3)
![Page 7: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/7.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 2: Finding the Coordinates of an Endpoint
Step 1 Let the coordinates of P equal (x, y).
Step 2 Use the Midpoint Formula.
P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q.
![Page 8: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/8.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 2 Continued
Multiply both sides by 2.
Isolate the variables.
–2 = –5 + x+5 +5
3 = x
6 = 4 + y
−4 −4
Simplify. 2 = y
Set the coordinates equal.
Step 3 Find the x-coordinate.
Find the y-coordinate.
Simplify.
![Page 9: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/9.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
The coordinates of Q are (3, 2).
N (–5, 4)P(–1, 3)
Q (3, 2)
Check Graph points Q and N and midpoint P.
Additional Example 2 Continued
![Page 10: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/10.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
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.
![Page 11: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/11.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 2 Continued
Multiply both sides by 2.
Simplify.
Simplify.
Isolate the variables.
–2 = –6 + x
+6 +6
2 = –1 + y
+1 +1
4 = x 3 = y
Set the coordinates equal.
Step 3 Find the x-coordinate.
Find the y-coordinate.
![Page 12: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/12.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
The coordinates of T are (4, 3)
R(–6, –1)
T(4, 3)
S(–1, 1)
Check Graph points R and S and midpoint T.
Check It Out! Example 2 Continued
![Page 13: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/13.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
You can also use coordinates to find the distance between two points or the length of a line segment. To find the length of segment PQ, draw a horizontal segment from P and a vertical segment from Q to form a right triangle.
![Page 14: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/14.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a2 + b2 = c2.
Remember!
![Page 15: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/15.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 3: Finding Distance in the Coordinate Plane
Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3).
Distance Formula
Substitute (4, –2) for (x1, y1) and (3, –2) for (x2, y2).
Subtract.
Simplify powers.
Add.
Find the square root to the nearest hundredth.
![Page 16: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/16.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 3 Continued
A (–2, –2)
B (4, 3)6
5
Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3).
![Page 17: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/17.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 3Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1).
Distance Formula
Substitute (3, 2) for (x1, y1) and (-3, -1) for (x2, y2).
Add.
Simplify powers.
Add.
Find the square root to the nearest hundredth.
![Page 18: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/18.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 3 Continued
S(–3, –1)
R(3, 2)63
Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1).
![Page 19: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/19.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Each unit on the map represents 100 meters. To the nearest tenth of a meter, how far is it from the roller coaster to the Ferris wheel?
Additional Example 4: Application
Substitute.
Add.
Simplify powers.
Find the square root to the nearest tenth.
It is 7.211 100 or 721.1 meters from the roller coaster to Ferris Wheel.
![Page 20: Apply the formula for midpoint. Use the distance formula to find the distance between two points.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56814839550346895db557d5/html5/thumbnails/20.jpg)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 4Jacob takes a boat from Pahokee to Clewiston. To the nearest tenth of a mile, how far does he travel?
d 17.7 miles
Substitute.
Square.
Simplify powers.
Find the square root to the nearest tenth.