Key Concepts
- Find segment lengths
- Use algebra with segment lengths
- Use the midpoint formula
Introduction
In this chapter, we will learn to find segment lengths based on the midpoint, using algebra to find segment lengths, using the midpoint formula and distance formula.
Midpoints
The midpoint of a segment is the point that divides the segment into two congruent segments.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (1) Midpoint and Distance Formulas - Explanation, Uses | Turito (1)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2141.png)
M is the midpoint of segment AC
M bisects segment AC
Bisectors
A segment bisector can be a point, ray, line, line segment, or plane that intersects the segment at its midpoint.
A midpoint or a segment bisector bisects a segment.
Example of segment bisectors:
![Midpoint and Distance Formulas - Explanation, Uses | Turito (3) Midpoint and Distance Formulas - Explanation, Uses | Turito (3)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2143.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (4) Midpoint and Distance Formulas - Explanation, Uses | Turito (4)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2146.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (5) Midpoint and Distance Formulas - Explanation, Uses | Turito (5)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2144.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (6) Midpoint and Distance Formulas - Explanation, Uses | Turito (6)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2145.png)
Find segment lengths
Example 1:
In the skateboard design, bisects
at point T, and
= 39.9 cm. Find
.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (11) Midpoint and Distance Formulas - Explanation, Uses | Turito (11)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2148.png)
Solution:
Point T is the midpoint of XY. So, XT = TY = 39.9cm.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (13) Midpoint and Distance Formulas - Explanation, Uses | Turito (13)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2371.png)
Example 2:
Find RS.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (14) Midpoint and Distance Formulas - Explanation, Uses | Turito (14)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2154.png)
Solution:
Point T is the midpoint of RS. So RT= TS = 21.7
![Midpoint and Distance Formulas - Explanation, Uses | Turito (15) Midpoint and Distance Formulas - Explanation, Uses | Turito (15)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2372.png)
Use algebra with segment lengths
Example 3:
Point M is the midpoint of VW. Find the length of VW .
![Midpoint and Distance Formulas - Explanation, Uses | Turito (16) Midpoint and Distance Formulas - Explanation, Uses | Turito (16)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2155.png)
Solution:
STEP 1: Write and solve an equation. Use the fact that VM = MW
![Midpoint and Distance Formulas - Explanation, Uses | Turito (17) Midpoint and Distance Formulas - Explanation, Uses | Turito (17)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2374.png)
Example 4:
Point C is the midpoint of BD. Find the length of BC.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (18) Midpoint and Distance Formulas - Explanation, Uses | Turito (18)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2163.png)
Solution:
Step 1: Write and solve an equation.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (19) Midpoint and Distance Formulas - Explanation, Uses | Turito (19)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2375.png)
Use the Mid Point Formula
Midpoint formula
![Midpoint and Distance Formulas - Explanation, Uses | Turito (20) Midpoint and Distance Formulas - Explanation, Uses | Turito (20)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2165.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (21) Midpoint and Distance Formulas - Explanation, Uses | Turito (21)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2164.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (22) Midpoint and Distance Formulas - Explanation, Uses | Turito (22)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2168.png)
Example 5:
- Find midpoint: The endpoints of RS are R(1, 23) and S(4, 2). Find the coordinates of the midpoint M.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (23) Midpoint and Distance Formulas - Explanation, Uses | Turito (23)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2167.png)
Solution:
- Find midpoint:
Use the midpoint formula.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (24) Midpoint and Distance Formulas - Explanation, Uses | Turito (24)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2172.png)
The coordinates of the midpoint M are:
![Midpoint and Distance Formulas - Explanation, Uses | Turito (25) Midpoint and Distance Formulas - Explanation, Uses | Turito (25)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2166.png)
- Find endpoint: The midpoint of JK−JK- is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (26) Midpoint and Distance Formulas - Explanation, Uses | Turito (26)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2169.png)
Solution:
- Find endpoint:
Let (x, y) be the coordinates of endpoint K.
Use the midpoint formula.
STEP 1: Find x.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (27) Midpoint and Distance Formulas - Explanation, Uses | Turito (27)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2170.png)
1+ x = 4
x = 3
STEP 2: Find y.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (28) Midpoint and Distance Formulas - Explanation, Uses | Turito (28)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2171.png)
4 + y = 2
Y = -2
The coordinates of endpoint K are (3, -2).
Distance Formula
![Midpoint and Distance Formulas - Explanation, Uses | Turito (29) Midpoint and Distance Formulas - Explanation, Uses | Turito (29)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2175.png)
The distance formula is a formula for computing the distance between two points in a coordinate plane.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (30) Midpoint and Distance Formulas - Explanation, Uses | Turito (30)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2173.png)
If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is
![Midpoint and Distance Formulas - Explanation, Uses | Turito (31) Midpoint and Distance Formulas - Explanation, Uses | Turito (31)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2173.png)
Example 6:
Find distance between R and S. Round to the nearest tenth if needed.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (32) Midpoint and Distance Formulas - Explanation, Uses | Turito (32)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2174.png)
![Midpoint and Distance Formulas - Explanation, Uses | Turito (33) Midpoint and Distance Formulas - Explanation, Uses | Turito (33)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/solution.png)
Exercise
- What is the difference between these three symbols: = and ≈?
- Identify the segment bisectors of Then find
![Midpoint and Distance Formulas - Explanation, Uses | Turito (34) Midpoint and Distance Formulas - Explanation, Uses | Turito (34)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2176.png)
- Identify the segment bisectors of RS. Then find RS
![Midpoint and Distance Formulas - Explanation, Uses | Turito (35) Midpoint and Distance Formulas - Explanation, Uses | Turito (35)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2177.png)
- Identify the segment bisectors of XY Then find XY
![Midpoint and Distance Formulas - Explanation, Uses | Turito (36) Midpoint and Distance Formulas - Explanation, Uses | Turito (36)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2178.png)
- Identify the segment bisectors of XYThen find XY
![Midpoint and Distance Formulas - Explanation, Uses | Turito (37) Midpoint and Distance Formulas - Explanation, Uses | Turito (37)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2179.png)
- What is the approximate length of ABwith endpoints A(-3, 2) and B(1, -4)?
- What is the approximate length of RS with endpoints R(2, 3) and S(4, -1)?
- Find the midpoint of the segment between (20, -14) and (-16, 4).
- The midpoint of segment DHis O(3, 4). One endpoint is D(5, 7). Find the coordinates of H.
![Midpoint and Distance Formulas - Explanation, Uses | Turito (38) Midpoint and Distance Formulas - Explanation, Uses | Turito (38)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2180.png)
- Work with a partner. Use centimeter graph paper.
- Graph AB , where the points A and B are as shown below.
- Explain how to bisect AB , that is, to divide AB into two congruent line segments. Then bisect AB and use the result to find the midpoint M of AB .
![Midpoint and Distance Formulas - Explanation, Uses | Turito (39) Midpoint and Distance Formulas - Explanation, Uses | Turito (39)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2183.png)
- What are the coordinates of the midpoint M?
- Compare the x-coordinates of A, B, and M. Compare the y-coordinates of A, B, and M. How are the coordinates of the midpoint M related to the coordinates of A and B?
What have we learned
- Finding segment lengths based on midpoint.
- Using algebra to find segment lengths.
- Using the midpoint formula and distance formula.
Concept Map
![Midpoint and Distance Formulas - Explanation, Uses | Turito (40) Midpoint and Distance Formulas - Explanation, Uses | Turito (40)](https://i0.wp.com/www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2186-1024x510.png)
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