Distance between two points pythagorean theorem worksheet answers

Video transcript

- We are asked what is the distance between the following points. Pause this video and see if you can figure it out. There's multiple ways to think about it. The way I think about it is really to try to draw a right triangle where these points, where the line that connects these points is the hypotenuse and then we can just use the Pythagorean Theorem. Let me show you what I am talking about. Let me draw a right triangle, here. That is the height of my right triangle and this is the width of my right triangle. Then the hypotenuse will connect these two points. I could use my little ruler tool here to connect that point and that point right over there. I'll color it in orange. There you have it. There you have it. I have a right triangle where the line that connects those two points is the hypotenuse of that right triangle. Why is that useful? From this, can you pause the video and figure out the length of that orange line, which is the distance between those two points? What is the length of this red line? You could see it on this grid, here. This is equal to two. It's exactly two spaces, and you could even think about it in terms of coordinates. The coordinate of this point up here is negative five comma eight. Negative five comma eight. The coordinate here is X is four, Y is six. Four comma six, and so the coordinate over here is going to have the same Y coordinate as this point. This is going to be comma six. It's going to have the same X coordinate as this point. This is going to be negative five comma six. Notice, you're only changing in the Y direction and you're changing by two. What's the length of this line? You could count it out, one, two, three, four, five, six, seven, eight, nine. It's nine, or you could even say hey look, we're only changing in the X value. We're going from negative five, X equals negative five, to X equals four. We're going to increase by nine. All of that just sets us up so that we can use the Pythagorean Theorem. If we call this C, we know that A squared plus B squared is equal to C squared, or we could say that two squared ... Let me do it over here. Use that same red color. Two squared plus nine squared, plus nine squared, is going to be equal to our hypotenuse square, which I'm just calling C, is going to be equal to C squared, which is really the distance. That's what we're trying to figure out. Two squared, that is four, plus nine squared is 81. That's going to be equal to C squared. We get C squared is equal to 85. C squared is equal to 85 or C is equal to the principal root of 85. Can I simplify that a little bit? Let's see. How many times does five go into 85? It goes, let's see, it goes 17 times. Neither of those are perfect squares. Yeah, that's 50 plus 35. Yeah, I think that's about as simple as I can write it. If you wanted to express it as a decimal, you could approximate it by putting this into a calculator and however precise you want your approximation to be. That over here, that's the length of this line, our hypotenuse and our right triangle, but more importantly for the question they're asking, the distance between those points.

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.

A General Note: The Distance Formula

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]

Example: Finding the Distance between Two Points

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

Try It

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.

Example: Finding the Distance between Two Locations

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.

Example: Finding the Midpoint of the Line Segment

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

Try It

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

Example: Finding the Center of a Circle

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.

Try It

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