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Distance Formula

Grade 10·21 problems·~30 min·ACT Math·topic·act-geo-coord-distance
Work through problems with immediate feedback
A

Recall / Warm-Up

1.

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

2.

Simplify 50\sqrt{50}.

3.

What is the distance between the points (2, 5) and (2, −3) on the coordinate plane?

B

Fluency Practice

1.

Find the distance between (1, 2) and (4, 6).

2.

Find the distance between (−3, 5) and (2, −7).

3.

What is the distance between (2, 1) and (5, 3)?

4.

Find the distance between (−4, −1) and (2, 7).

5.

Find the distance between (0, 0) and (1, 1). Express in simplified radical form.

C

Varied Practice

Triangle ABC with vertices A(0,0), B(6,0), C(3,4) plotted on a coordinate grid.
1.

Triangle ABCABC has vertices A(0,0)A(0, 0), B(6,0)B(6, 0), and C(3,4)C(3, 4). What is the perimeter of triangle ABCABC?

Triangle PQR with vertices P(0,0), Q(4,0), R(0,3) showing the right angle at P.
2.

A triangle has vertices P(0,0)P(0, 0), Q(4,0)Q(4, 0), and R(0,3)R(0, 3). Which classification best describes triangle PQRPQR?

3.

Is the point (3,4)(3, 4) equidistant from (0,0)(0, 0) and (6,0)(6, 0)?

4.

The distance between (1,2)(1, 2) and (x,2)(x, 2) is 7. The possible values of xx are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

x values:
5.

The distance between (3,y)(3, y) and (3,5)(3, 5) is 9. Which of the following lists all possible values of yy?

D

Word Problems / Application

1.

Three cell towers are located at A(0,0)A(0, 0), B(3,4)B(3, 4), and C(8,1)C(8, 1). Which pair of towers is closest to each other?

Rectangular park ABCD with vertices at (0,0), (8,0), (8,6), (0,6) and diagonal AC shown.
2.

A park has four corners at A(0,0)A(0, 0), B(8,0)B(8, 0), C(8,6)C(8, 6), and D(0,6)D(0, 6). A diagonal path runs from AA to CC.

1.

What is the perimeter of the park?

2.

What is the length of the diagonal path from A to C?

3.

On a map, a school is at (2,3)(2, 3) and a library is at (−1,7)(−1, 7). Each grid unit represents 1 mile.

What is the straight-line distance between the school and the library?

E

Error Analysis

1.

Maya solved this problem:

"Find the distance between (1, 3) and (4, 7)."

Maya's work:

  1. (4−1)+(7−3)=3+4=7(4 - 1) + (7 - 3) = 3 + 4 = 7
  2. d=7≈2.65d = \sqrt{7} \approx 2.65

What error did Maya make, and what is the correct distance?

2.

Jake solved this problem:

"Find the distance between (−2, 3) and (4, 7)."

Jake's work:

  1. Horizontal: (4−(−2))2=36=6\sqrt{(4 - (-2))^2} = \sqrt{36} = 6
  2. Vertical: (7−3)2=16=4\sqrt{(7 - 3)^2} = \sqrt{16} = 4
  3. d=6+4=10d = 6 + 4 = 10

What error did Jake make, and what is the correct distance?

F

Challenge / Extension

1.

The distance between (2,y)(2, y) and (5,1)(5, 1) is 5. The possible values of yy are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

y values:
2.

Find the distance between the points (1,2,3)(1, 2, 3) and (4,6,6)(4, 6, 6) in three-dimensional space.

0 of 21 answered