Midpoint and Distance Formulas - Explanation, Uses | Turito (2024)

MathMidpoint and Distance Formulas

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.

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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:

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Find segment lengths

Example 1:

In the skateboard design, Midpoint and Distance Formulas - Explanation, Uses | Turito (7) bisects Midpoint and Distance Formulas - Explanation, Uses | Turito (8)at point T, and Midpoint and Distance Formulas - Explanation, Uses | Turito (9)= 39.9 cm. Find Midpoint and Distance Formulas - Explanation, Uses | Turito (10).

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Solution:

Point T is the midpoint of XY. So, XT = TY = 39.9cm.

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Example 2:

Find RS.

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Solution:

Point T is the midpoint of RS. So RT= TS = 21.7

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Use algebra with segment lengths

Example 3:

Point M is the midpoint of VW. Find the length of VW .

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Solution:

STEP 1: Write and solve an equation. Use the fact that VM = MW

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Example 4:

Point C is the midpoint of BD. Find the length of BC.

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Solution:

Step 1: Write and solve an equation.

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Use the Mid Point Formula

Midpoint formula

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Example 5:

  • Find midpoint: The endpoints of RS are R(1, 23) and S(4, 2). Find the coordinates of the midpoint M.
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Solution:

  • Find midpoint:

Use the midpoint formula.

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The coordinates of the midpoint M are:

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  • Find endpoint: The midpoint of JK−JK- is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
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Solution:

  • Find endpoint:

Let (x, y) be the coordinates of endpoint K.

Use the midpoint formula.

STEP 1: Find x.

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1+ x = 4

x = 3

STEP 2: Find y.

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4 + y = 2

Y = -2

The coordinates of endpoint K are (3, -2).

Distance Formula

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The distance formula is a formula for computing the distance between two points in a coordinate plane.

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If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is

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Example 6:

Find distance between R and S. Round to the nearest tenth if needed.

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Midpoint and Distance Formulas - Explanation, Uses | Turito (33)

Exercise

  • What is the difference between these three symbols: = and ≈?
  • Identify the segment bisectors of Then find
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  • Identify the segment bisectors of RS. Then find RS
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  • Identify the segment bisectors of XY Then find XY
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  • Identify the segment bisectors of XYThen find XY
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  • 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.
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  • 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 .
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  • 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

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Comments:

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FAQs

What is the midpoint formula and what is it used for? ›

Midpoint formula is a mathematical equation that is used to locate the halfway point between two data points. Besides in geometry, the study of economics uses this calculation to find the coefficient of elasticity, etc.

How to use midpoint and distance formula? ›

The midpoint formula is M = ( x 1 + x 2 2 , y 1 + y 2 2 ) ), where M is the midpoint and the other terms have the same definitions as they do in the distance formula. The slope equation is m = y 2 − y 1 x 2 − x 1 , where m is the slope and the other terms have the same definitions as they do in the distance formula.

What is the importance of midpoint formula in real life? ›

Finding the midpoint helps calculate geographical, computer programing, and economic problems in the real world. The midpoint formula in coordinate geometry is an equation that calculates the halfway point distance between two known coordinate points.

What are the learning objectives of midpoint and distance formula? ›

Learning Objectives

Find the distance between two points in the coordinate plane. Find the missing coordinate of a point given the distance from another known point. Find the midpoint of a line segment. Solve real-world problems using distance and midpoint formulas.

What is the distance formula used for? ›

distance formula, Algebraic expression that gives the distances between pairs of points in terms of their coordinates (see coordinate system). In two- and three-dimensional Euclidean space, the distance formulas for points in rectangular coordinates are based on the Pythagorean theorem.

What is the main reason for using the midpoint method? ›

The midpoint formula is used to calculate elasticity because it gives us the same elasticity value regardless of if the price increases or decreases, whereas when using the point elasticity we have to know which value is the initial value.

What is the midpoint rule for finding area? ›

The midpoint rule approximates this area by the area of a rectangle of width xj−xj−1=Δx x j − x j − 1 = Δ x and height f(¯xj) f ( x ¯ j ) which is the exact height at the midpoint of the range covered by x. x . ∫xjxj−1f(x)dx≈f(¯xj)Δx.

Can we use midpoint formula for equidistant? ›

Midpoint Formula

Another way to find an equidistant point is when it is the midpoint of a line segment. A midpoint is a point located at the middle of two connected points forming a line. With the coordinates of two given points, the equidistant point can be found using the midpoint formula.

When should we use midpoint formula? ›

Midpoint formula is used to find the centre point of a straight line. Sometimes you will need to find the number that is half of two particular numbers. For that, you find the average of the two numbers.

What is the advantage of using the midpoint formula? ›

The advantage of the midpoint method is that one obtains the same elasticity between two price points whether there is a price increase or decrease. This is because the formula uses the same base for both cases.

Which situation would use the midpoint formula? ›

The midpoint formula is appropriately used when one needs to find the middle or median point between two distinct points, for example, when trying to hang a lamp in the center of a room. However, it is not used in calculating a linear distance, such as the length of fencing needed or the distance to school.

What are the real life applications of distance formula? ›

It is used in navigation. The pilot of a plane calculates the distance between their plane and the other plane using the distance formula. They find the coordinate of the plane and then apply the distance formula to get the distance.

What does the midpoint formula help you find? ›

The midpoint formula is used to find the midpoint between two points whose coordinates are known to us. The midpoint formula is also used to find the coordinates of the endpoint if we know the coordinates of the other endpoint and the midpoint.

Why is it important to study the distance formula? ›

The distance formula can be used to determine the length of material needed for a building project. The distance formula can be used to determine the length of the sides of a shape.

What is the midpoint theorem used for? ›

The theory of midpoint theorem is used in coordinate geometry, stating that the midpoint of the line segment is an average of the endpoints. The 'x' and the 'y' coordinates must be known for solving an equation using this theorem. The Mid- Point Theorem is also useful in the fields of calculus and algebra.

What is midpoint purpose? ›

A story's midpoint is a moment when characters face higher stakes, significant changes, greater challenges, or story-altering revelations that propel the story into a new direction.

What is midpoint used for in statistics? ›

In conclusion, finding the midpoint in statistics involves averaging the two extremes of a data set or interval, offering a straightforward way to identify the central point. The skill of standing out online is needed more than ever in today's digital world.

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