Teaching Notes 4.16: Using the Distance Formula to Find
the Distance Between Two Points
Many students find the distance formula,d=√
(x 2 −x 1 )^2 +(y 2 −y 1 )^2 , intimidating because of
the subscripts, squares, and square root. A fundamental understanding of this notation enables
most students to use this formula correctly.- Refer to the diagram on the worksheet, noting that the distance between (1, 4) and (2, 3)
is the length of the hypotenuse of the right triangle. Note that the length of each leg is
one unit. Explain that the Pythagorean theorem states that in a right trianglea^2 +b^2 =c^2
wherecis the hypotenuse. Becausea=1andb=1, 1^2 + 12 =c^2 , thereforec=
√
2, which
is about 1.414.- Explain that it is inefficient to make a diagram and use the Pythagorean theorem to find the
distance between two points. The distance formula may be used instead. - Present the distance formula along with a coordinate plane on which the points (1, 4) and
(2, 3) are plotted. (These points also serve as the example on the worksheet.) - Explain the meaning of each variable in the distance formula.drepresents the distance
between the points.x 2 represents the value of thex-coordinate in the second ordered pair.
x 1 represents the value of thex-coordinate in the first ordered pair. Similarly,y 2 andy 1
represent the values of they-coordinate in the second and first ordered pair, respectively. - Review the information and example on the worksheet with your students. Explain the steps
for using the distance formula to find the distance between (1, 4) and (2, 3) as shown in
the example. Depending on their abilities, you may find it necessary to review the order of
operations and rounding to the nearest hundredth. See 1.5: ‘‘Simplifying Expressions with
Grouping Symbols and Exponents.’’
EXTRA HELP:
Simplify the expression inside the radical symbol first. Then find the square root.ANSWER KEY:
(1)√
29 ≈ 5. 39 (2)
√
20 ≈ 4. 47 (3)
√
34 ≈ 5. 83 (4)
√
13 ≈ 3. 61
(5)
√
8 ≈ 2. 83 (6) 10 (7)
√
17 ≈ 4. 12 (8)
√
5 ≈ 2. 24
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(Challenge)Disagree. Sammie’s method is the shortest way to find the distance between two
points on a vertical line. However, she could have also used the distance formula to obtain the
same answer.
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