how to find the length of a segment

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You can measure the length of a vertical or horizontal line on a coordinate plane by simply counting coordinates; however, measuring the length of a diagonal line is trickier. You can use the Distance Formula to find the length of such a line. This formula is basically the Pythagorean Theorem, which you can see if you imagine the given line segment as the hypotenuse of a right triangle.^[1] By using a basic geometric formula, measuring lines on a coordinate path becomes a relatively easy task.

1

Set up the Distance Formula. The formula states that $d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}$ , where $d$ equals the distance of the line, $(x_{1},y_{1})$ equal the coordinates of the first endpoint of the line segment, and $(x_{2},y_{2})$ equal the coordinates of the second endpoint of the line segment.^[2]
2

Find the coordinates of the line segment's endpoints. These might already be given. If not, count along the x-axis and y-axis to find the coordinates.^[3]

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3

Plug the coordinates into the Distance Formula. Be careful to substitute the values for the correct variables. The two $x$ coordinates should be inside the first set of parentheses, and the two $y$ coordinates should be inside the second set of parentheses.^[4]

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1

Calculate the subtraction in parentheses. By using the order of operations, any calculations in parentheses must be completed first.^[5]
2

Square the value in parentheses. The order of operations states that exponents should be addressed next.^[6]
3

Add the numbers under the radical sign. You do this calculation as if you were working with whole numbers.
4

Solve for $d$ d {\displaystyle d} . To reach your final answer, find the square root of the sum under the radical sign.

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Question

What do we call points that are on same line?

Points on the same straight line are said to be "collinear" or "colinear."
Question

How do I simplify the square root?

Look for a perfect square inside the radical sign, find its square root, and put that square root out in front of the radical sign, indicating that it's to be multiplied by the radical. For example: √50 = √(2 x 25) = 5√2.
Question

The rise is 4 inches. The angle is 90 degrees from the height to the base. What is the slope?

To find the slope, you have to know the horizontal distance over which the rise occurs. The 90° angle is not relevant.
Question

How do I find the distance of a line if I don't know the end points?

Assuming you don't have a measuring tool (tape measure, ruler, etc.), you cannot use the distance formula outlined above without knowing the end points. However, if the line is given as part of a geometrical figure, there are various geometrical and trigonometrical methods to find the length (too numerous and complicated to treat here).
Question

How would I find the length of each segment?

For a line segment, you should create a right triangle by making two more lines. If this line segment is on a dot grid, every line between two dots is one unit. If the line segment is on a coordinate plane, then you can do the same or use the distance formula for a line.
Question

A square table has an area of 16 sq cm. How do I find the length of its side?

√16 = 4 cm.
Question

On a number line, the length of line segment joining 3 and -3 is what?

On a number line 3 and -3 are each 3 units from zero (in opposite directions). So the total distance between them is 3 + 3 = 6.
Question

Would line AB be the same as line BA?

Yes.
Question

I don't understand the distance formula. Can you help?

It's simply the Pythagorean theorem. Find the vertical change (between endpoints of the line). Square it. Find the horizontal change. Square that, too. Add together the two squares. Then find the square root of that sum. This gives you the line length you're looking for.
Question

If I know the length of the line and one of the endpoints, how do I find the other endpoint?

You can't do it unless you also know the slope of the line.

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Remember the order of operations when calculating your answer. Subtract first, then square the differences, then add, and then find the square root.
Do not confuse this formula with others, like the Midpoint Formula, Slope Formula, Equation of a Line or Line Formula.

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To use the distance formula to find the length of a line, start by finding the coordinates of the line segment's endpoints. Then, plug the coordinates into the distance formula. Next, subtract the numbers in parenthesis and then square the differences. Once you've done that, just add the numbers that are under the radical sign and solve for d. To learn how to set up the distance formula, keep reading!

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