The distance formula

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   AB = DC = √29, BC = AD = 5, AC = 10, BD = 2√2. Milo's quilt has corners A(0, 0), B(5, 2), C(8, 6) and D(3, 4). Which set of facts shows ABCD is a parallelogram?

   • AC = 10 and BD = 2√2, so the diagonals are different
   • AB = √29 and BC = 5 differ, so ABCD is not a parallelogram
   • AB = DC = √29, because one pair of equal sides is enough
   • AB = DC = √29 and BC = AD = 5, so both pairs of opposite sides are equal
2. 2.

   Bo proves the diagonals of every rectangle are equal, using corners A(0, 0), B(w, 0), C(w, h) and D(0, h). Which proof works?

   • The diagonals bisect each other, so they are equal
   • AC = w + h and BD = w + h, so they match
   • With w = 6 and h = 4, AC = BD = √52, so it is true for all rectangles
   • AC = √(w² + h²) and BD = √(w² + h²), for every w and h
3. 3.

   Points A(1, 2, 3) and B(4, 6, 15). Each point is (x, y, height). Differences: 3, 4 and 12. Jun's drone flies from A(1, 2, 3) to B(4, 6, 15) in space, measured in meters. In a straight line, how far is that?

   • 5 m
   • 19 m
   • 13 m
   • 169 m
4. 4.

   Kira's circle has diameter AB with A(-2, 1) and B(6, 7). What are its center and radius?

   • Center (2, 4), radius 25
   • Center (2, 4), radius 5
   • Center (2, 4), radius 10
   • Center (4, 8), radius 5
5. 5.

   Point (x, 2x) with distance 10 from the origin (0, 0). Esme's point (x, 2x) lies 10 units from the origin. What are the possible values of x?

   • x = 2√5 or x = -2√5
   • x = 5√2 or x = -5√2
   • x = 20 or x = -20
   • x = 10 or x = -10
6. 6.

   Noah hikes from A(0, 0) to B(6, 8), then on to C(12, 0), in km. How much farther is that than walking straight from A to C?

   • 12 km
   • 16 km
   • 8 km
   • 20 km
7. 7.

   Pia's radio tower is at (3, 4). Its signal reaches 5 km, and the road is the x-axis. Where does the signal's edge meet the road?

   • (0, 0) alone
   • (3, 0) alone
   • (-2, 0) and (8, 0)
   • (0, 0) and (6, 0)
8. 8.

   Points (-3, 2) and (5, -4). Zee wrote √((5 − 3)² + (-4 − 2)²). Zee finds the distance from (-3, 2) to (5, -4) as √((5 − 3)² + (-4 − 2)²) = √40. What is wrong, and what is the right distance?

   • The rise should be -4 + 2 = -2; the distance is √8
   • Nothing is wrong; √40 is right
   • The legs should be added: 2 + 6 = 8
   • The run is 5 − (-3) = 8, not 2; the distance is 10
9. 9.

   Luz compares three drones P(5, 4), Q(-2, 5) and R(5, -4) from a base at B(1, 1). Which statement is correct?

   • Q is farthest, since its x-gap is the largest
   • P, Q and R are each 5 away from the base
   • P is farthest, since it is listed first
   • R is farthest at √41; P and Q tie at 5

10. Read this first

    Use this

    A(1, 0) and B(5, 4). A point (x, y) is equally far from both when its distance to A equals its distance to B.

11. 10.

    Ren's points A(1, 0) and B(5, 4) are the ends of a path. Which equation describes every point equally far from A and B?

    • x + y = 5
    • y = x − 1
    • x + y = 10
    • x + y = 3