What is the distance between points A 7 5 and B 2 5?

Find the distance between two points from x and y coordinates with this distance formula calculator.

Distance Formula :

Example:

For two points, (3,2) and (15, 10) the distance is calculated as:

Distance = 14.42 (rounded to the nearest 100th)

More common examples to try with the Distance Formula Calculator.

What is the distance between points f(2, 9) and g(4, 14)? If you apply the distance formula, or insert it into the calculator, you'd find an answer of 5.39 units. If you round this to the nearest whole number, this would be 5.

What is the distance between points f(6, 4) and g(14, 19)? Using the distance formula calculator, this gives an even result of 17 units.

Try two points with one as a negative, f(2, 9) and g(-2, 6)? Using the distance formula calculator, this gives an even result of 5 units.

What is the distance between points f(3, 12) and g(14, 2)? The answer is 14.87 units.

What is the distance between points f(13, 2) and g(7, 10)? The result is 10 units.

Try these coordinates with some negatives,(-6, -1) and g(4, 3). Using the distance formula calculator, this gives a result of 10.77 units.

What is the distance between points f(-5, -2) and g(2, 4)? Using the distance formula calculator, this gives a result of 9.22 units.

Learning Outcomes

• Use the distance formula to find the distance between two points in the plane.
• Use the midpoint formula to find the midpoint between two points.

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[/latex], is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.

The relationship of sides [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.

[latex]{c}^{2}={a}^{2}+{b}^{2}\rightarrow c=\sqrt{{a}^{2}+{b}^{2}}[/latex]

It follows that the distance formula is given as

[latex]{d}^{2}={\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}\to d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}[/latex]

We do not have to use the absolute value symbols in this definition because any number squared is positive.

Given endpoints [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the distance between two points is given by

[latex]d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}[/latex]

Find the distance between the points [latex]\left(-3,-1\right)[/latex] and [latex]\left(2,3\right)[/latex].

Find the distance between two points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,9\right)[/latex].

In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.

Let’s return to the situation introduced at the beginning of this section.

Tracie set out from Elmhurst, IL to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.

Using the Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment, [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[/latex].

[latex]M=\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)[/latex]

A graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.

Find the midpoint of the line segment with the endpoints [latex]\left(7,-2\right)[/latex] and [latex]\left(9,5\right)[/latex].

Find the midpoint of the line segment with endpoints [latex]\left(-2,-1\right)[/latex] and [latex]\left(-8,6\right)[/latex].

The diameter of a circle has endpoints [latex]\left(-1,-4\right)[/latex] and [latex]\left(5,-4\right)[/latex]. Find the center of the circle.

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You know that the distance A B between two points in a plane with Cartesian coordinates A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is given by the following formula:

A B = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2

The distance formula is really just the Pythagorean Theorem in disguise.

To calculate the distance A B between point A ( x 1 , y 1 ) and B ( x 2 , y 2 ) , first draw a right triangle which has the segment A B ¯ as its hypotenuse.

If the lengths of the sides are a and b , then by the Pythagorean Theorem,

( A B ) 2 = ( A C ) 2 + ( B C ) 2

Solving for the distance A B , we have:

A B = ( A C ) 2 + ( B C ) 2

Since A C is a horizontal distance, it is just the difference between the x -coordinates: | ( x 2 − x 1 ) | . Similarly, B C is the vertical distance | ( y 2 − y 1 ) | .

Since we're squaring these distances anyway (and squares are always non-negative), we don't need to worry about those absolute value signs.

A B = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2

Example:

Find the distance between points A and B in the figure above.

In the above example, we have:

A ( x 1 , y 1 ) = ( − 1 , 0 ) , B ( x 2 , y 2 ) = ( 2 , 7 )

so

A B = ( 2 − ( − 1 ) ) 2 + ( 7 − 0 ) 2                 = 3 2 + 7 2                 = 9 + 49                   = 58

or approximately 7.6 units.