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?

   • 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
   • AC = 10 and BD = 2√2, so the diagonals are different
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?

   • 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
   • The diagonals bisect each other, so they are equal
   • AC = w + h and BD = w + h, so they match
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?

   • 169 m
   • 5 m
   • 19 m
   • 13 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 = 5√2 or x = -5√2
   • x = 2√5 or x = -2√5
   • x = 10 or x = -10
   • x = 20 or x = -20
6. 6.

   Tala's sail has corners A(0, 0), B(5, 1) and C(2, 4), with AB² = 26, BC² = 18 and AC² = 20. What kind of angle is the largest angle?

   • Right, since the triangle is isosceles: 18 and 20 are close
   • Obtuse, since AB² = 26 is bigger than both 18 and 20
   • Right, since 18 + 20 is close to 26
   • Acute: 26 is less than 18 + 20 = 38, so the angle at C is acute
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?

   • (3, 0) alone
   • (0, 0) and (6, 0)
   • (-2, 0) and (8, 0)
   • (0, 0) alone
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 legs should be added: 2 + 6 = 8
   • The rise should be -4 + 2 = -2; the distance is √8
   • Nothing is wrong; √40 is right
   • 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?

   • P is farthest, since it is listed first
   • P, Q and R are each 5 away from the base
   • R is farthest at √41; P and Q tie at 5
   • Q is farthest, since its x-gap is the largest
10. 10.

    Kira's sail has corners (0, 0), (8, 0) and (4, k) with k > 0. The sail is equilateral. What is k?

    • √80
    • 2√3
    • 48
    • 4√3