Pythagorean Theorem To Find Distance

Learning Outcomes

Use the distance formula to observe the distance between two points in the airplane.
Employ the midpoint formula to discover the midpoint between ii points.

Derived from the Pythagorean Theorem, the altitude formula is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{ii}={c}^{2}[/latex], is based on a correct 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}_{ii}-{10}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] to side d is the same every bit that of sides a and b to side c. We use the accented value symbol to indicate that the length is a positive number because the accented value of any number is positive. (For example, [latex]|-iii|=iii[/latex]. ) The symbols [latex]|{10}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{one}|[/latex] indicate that the lengths of the sides of the triangle are positive. To detect 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}^{two}+{b}^{2}}[/latex]

It follows that the distance formula is given every bit

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

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

A General Note: The Altitude Formula

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

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

Example: Finding the Altitude betwixt Two Points

Observe the distance between the points [latex]\left(-3,-i\correct)[/latex] and [latex]\left(2,iii\correct)[/latex].

Try It

Observe the distance between ii points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,ix\right)[/latex].

Show Prove Solution

[latex]\sqrt{125}=v\sqrt{5}[/latex]

In the following video, nosotros present more worked examples of how to use the distance formula to find the distance betwixt two points in the coordinate aeroplane.

Example: Finding the Altitude between 2 Locations

Let's return to the situation introduced at the commencement of this department.

Tracie set out from Elmhurst, IL to get to Franklin Park. On the way, she made a few stops to do errands. Each end is indicated by a red dot. Discover the total distance that Tracie traveled. Compare this with the altitude betwixt her starting and terminal positions.

Using the Midpoint Formula

When the endpoints of a line segment are known, we tin can notice 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({ten}_{1},{y}_{one}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]1000[/latex].

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

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

This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.

Example: Finding the Midpoint of the Line Segment

Discover the midpoint of the line segment with the endpoints [latex]\left(seven,-2\right)[/latex] and [latex]\left(9,five\correct)[/latex].

Try It

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

Bear witness Solution

[latex]\left(-5,\frac{5}{2}\right)[/latex]

Example: Finding the Eye of a Circle

The bore of a circumvolve has endpoints [latex]\left(-1,-iv\correct)[/latex] and [latex]\left(5,-iv\right)[/latex]. Find the centre of the circumvolve.