Applying The Distance And Midpoint Formulas
The distance and midpoint formulas can help us find the distance and midpoint between two endpoints on a coordinate plane. The midpoint of a line segment formula between two coordinate pairs and
The distance of that same line segment denoted by coordinate pairs and
With a little more practice, you can navigate these formulas with ease.
Presentation On Theme: Lesson 811 Distance And Midpoint Formula Presentation Transcript:
1 Lesson 8.11 distance and midpoint formulaLesson 1-3: Formulas
2 Homework 1. P and R 2. S and TUse the Distance Formula to find the distance of:1. P and R2. S and TUse the Midpoint formula to find the midpoint between:3. W and X4. B and DLesson 1-3: Formulas
3 The Distance FormulaThe distance d between any two points with coordinates and is given by the formula d =Lesson 1-3: Formulas
4 Example 1: Finding Distances in the Coordinate PlaneUse the Distance Formula to find the distance, to the nearest tenth, fromD to E.
5 Example 1 ContinuedUse the Distance Formula. Substitute thevalues for the coordinates of D and E into theDistance Formula.
6 Example 2: Using the Distance FormulaFind the length for FG and JK.Then determine whether FG JK.Step 1 Find the coordinates of each point.F, G, J, K
7 Example 2 ContinuedStep 2 Use the Distance Formula.
8 Check It Out! Example 3Find EF and GH. Then determine if EF GH.Step 1 Find the coordinates of each point.E, F, G, H
9 Check It Out! Example 3 ContinuedStep 2 Use the Distance Formula.
10 Example 4: Sports ApplicationA player throws the ball from first base to a point located between third base and home plate and 10 feet from third base.What is the distance of the throw, to the nearest tenth?
13 Example 1: Finding the Coordinates of a MidpointFind the coordinates of the midpoint of PQ with endpoints P and Q.=
14 Check It Out! Example 1Find the coordinates of the midpoint of EF with endpoints E and F.
Unit 1 Geometry Basics Homework 3 Distance And Midpoint Formulas Answer Key
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Directions: Find the distance between each pair of points.1. 1-4.6) and 3. and
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How To Solve The Midpoint Formula
Youll use the midpoint formula to find the middle point between any two points. To do this, the formula looks at the coordinates of the endpoints and then finds the average value between the x-coordinates and the y-coordinates.
In the graph below, we have a line segment between the two coordinate points. is , and
The midpoint of the line segment is expressed as:
The red dot in the middle represents the new coordinate pair for the midpoint:
Lets substitute the coordinate values from above into the equation:
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The Distance And Midpoint Formulas
The distance formula is used to find the distance between two points in the coordinate plane. We’ll explain this using an example below
We want to calculate the distance between the two points and . We could see the line drawn between these two points is the hypotenuse of a right triangle. The legs of this triangle would be parallel to the axes which mean that we can measure the length of the legs easily.
We’ll get the length of the distance d by using the Pythagorean Theorem
$$d^=2^+6^=4+36=40$$
$$d=\sqrt\approx 6.32$$
This method can be used to determine the distance between any two points in a coordinate plane and is summarized in the distance formula
$$d=\sqrt-x_ \right )^+\left ^}$$
The point that is at the same distance from two points A and B on a line is called the midpoint. You calculate the midpoint using the midpoint formula
$$m =\left ,\: \: \left $$
We can use the example above to illustrate this
$$ m =\left ,\: \: \left =$$
$$=\left ,\: \: \left =\begin 1,\: 2 \end$$
How To Use The Distance And Midpoint Formulas On A Coordinate Plane
With the distance and midpoint formulas, you can find the distance and midpoint between any two points on a coordinate plane.
The distance formula gives you the distance d, expressed as a single value, between the two endpoints:
The midpoint formula gives you the midpoint, expressed as an ordered pair, between the two endpoints:
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How To Solve The Distance Formula
The distance formula works for any two given points. Let’s take a look at this example with coordinate points and :
If we imagine the two end points as two vertices on a triangle, we can see how the distance formula works. Remember that the hypotenuse of a right triangle, when squared, equals the sum of the square of the two legs. This is where the distance formula comes from.
Heres where the triangle is in our diagram:
In this example, the green and blue dotted sides of the triangle are perpendicular lines that form a right triangle.
This blue side of the triangle is 4 units long, and the green side of the triangle is 2 units long, as we can see on the graph. But how long is the red side? Its the distance between our two original points.
Lets apply the Pythagorean theorem, c² = a² + b². Well substitute the red line for c, the hypotenuse, and the green and blue lines for sides a and b.
Well take the square root of each side:
Finally, we substitute the colors. Red becomes D, or the distance.
Now, look at our coordinate pairs. Well refer to as and as .
Green is the change in the x-values, so well subtract the x-values of the two coordinate pairs,.
Likewise, the value for blue, the change in y, is .
The distance formula emerges when we substitute these values:
The length of the hypotenuse here is the distance between our end points. Lets substitute the points to figure out the answer:
Then, we simplify according to the order of operations, PEMDAS:
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