The distance formula

Grade 10 Math Worksheet - Building Confidence

Circle the best answer.

1. 1.

   Ari's map has the library at A(-2, 3) and the park at B(4, -5). Which expression gives the straight-line distance AB?

   • (4 − (-2)) + (3 − (-5))
   • √((4 − (-2)) + (-5 − 3))
   • √((4 − 2)² + (-5 − 3)²)
   • √((4 − (-2))² + (-5 − 3)²)
2. 2.

   Bo's kite string runs from D(1, -2) to E(13, 3) on a grid in meters. How long is the string?

   • 13 m
   • 169 m
   • √145 m
   • 17 m
3. 3.

   Jun's robot goes from (0, 0) to (5, 5). Which is its straight-line distance?

   • √10
   • 5√2
   • 10
   • 50
4. 4.

   Luz's quilt triangle has corners A(0, 0), B(4, 3) and C(8, 0). Is it isosceles?

   • No: AB = 5, BC = 7 and AC = 8, so the three differ
   • Yes: AC = 8 is the longest side, so the other two must be equal
   • Yes: AB = √(16 + 9) = 5 and BC = 5, while AC = 8
   • No: the base AC = 8 is longer, so the sides cannot match
5. 5.

   Milo's sail has corners P(0, 0), Q(2, 6) and R(8, 4), with PQ² = 40, QR² = 40 and PR² = 80. Is the angle at Q a right angle?

   • Yes: PQ + QR is greater than PR
   • Yes: PQ² + QR² = 40 + 40 = 80 = PR²
   • It cannot be told without measuring the angle
   • No: PQ = QR, so the angle at Q is acute
6. 6.

   A(1, 2) and B(x, 6) with AB = 5 and x > 0. Noah's point A is (1, 2) and B is (x, 6) with x positive. AB = 5. Which x works?

   • x = 3
   • x = 6
   • x = 4
   • x = 10
7. 7.

   A triangle with corners at known coordinates. Pia wants to show a triangle on the grid is isosceles. Which check works?

   • Find all three side lengths with the distance formula and see if two are equal
   • Check that two corners have the same y-coordinate
   • Find the midpoint of each side and compare them
   • Find the three slopes and see if two are equal
8. 8.

   Ren asks why the distance formula uses squares and a root. Which explanation is correct?

   • The run and rise are the legs of a right triangle, and the distance is its long side
   • The squared total is the distance, because squares measure space
   • The run and rise are added because distances add up
   • Just the longer of the run and rise matters
9. 9.

   Tala finds AB, then wonders whether swapping the points to find BA changes the answer. What is true?

   • AB = BA if the points lie on a horizontal line, but not otherwise
   • BA is the negative of AB, since the subtraction flips sign
   • AB = BA if both points have positive coordinates, but not otherwise
   • AB = BA, because (x₂ − x₁)² = (x₁ − x₂)²
10. 10.

    Zee's circle has center O(-2, 3) and passes through P(1, -1). What is its radius?

    • 7
    • 5
    • √17
    • 25