Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Two points on a coordinate grid that do not line up horizontally or vertically are joined by a slanted segment, and a ruler is not much help. The trick is to see that segment as the hypotenuse of a right triangle whose legs run along the grid lines. For the points (1, 2) and (7, 10), the horizontal leg is 7 - 1 = 6 units and the vertical leg is 10 - 2 = 8 units, so the distance is √(6² + 8²) = √100 = 10 units.
Grade 8 students find these legs by counting or subtracting coordinates, taking care when points sit in different quadrants: from x = -3 to x = 5 is 8 units, not 2. Because the legs are squared, the order of subtraction does not matter, and a negative difference gives the same result. This standard prepares the ground for the distance formula in high school, but at this level the emphasis is on drawing the right triangle and understanding why the method works, which also helps with tasks like finding the perimeter of a shape plotted on a grid.
From -3 to 5 students often say 2 units. Counting on the grid, or computing 5 - (-3), gives 8.
The distance from (0, 0) to (3, 4) is not 3 + 4 = 7. That is the route along the grid, not the straight-line distance, which is 5.
Students subtract an x-coordinate from a y-coordinate. Keep the horizontal change (x values) and vertical change (y values) separate.
Reporting 100 instead of 10 is common. The sum of squares is the square of the distance, so take its root.
Find the distance between A(-3, 1) and B(5, 7).
Answer: The distance from A to B is 10 units.
Have students plot the two points, draw the slope triangle with a colored pencil, and write the leg lengths on it before any calculation. A map task, such as distances between landmarks on a town grid, makes the method purposeful and shows the difference between a walking route along streets and a straight-line distance.
Assessment items usually show points on a grid or give coordinates and ask for the distance, sometimes to the nearest tenth. Encourage drawing even when coordinates are given, because the picture prevents sign errors.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 13
The legs are 5 and 12. √(25 + 144) = √169 = 13.
Answer: 5
Horizontal leg 3, vertical leg 4. √(9 + 16) = 5.
Answer: D) 10
6 - (-4) = 10. Lengths are always positive.
Answer: 3.6
Legs 3 and 2: √(9 + 4) = √13 ≈ 3.6.
Answer: B) Because the differences are squared, so a negative result becomes positive
Squaring a negative difference gives the same value as squaring the positive one, so the distance is unchanged.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.^*
A full lesson with slides, activities and an exit ticket on distance between points on a grid, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.B.8 with an answer key, ready in about a minute.
Make a worksheet →Turn distance between points on a grid into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It covers the same idea, but grade 8 students are expected to use the Pythagorean Theorem with a drawn triangle. The general distance formula is formalized in high school.
Yes. Points can be in any quadrant, so students need to find differences across zero carefully.