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 = DC = √29 and BC = AD = 5, so both pairs of opposite sides are equal
   • AB = DC = √29, because one pair of equal sides is enough
   • AB = √29 and BC = 5 differ, so ABCD is not a parallelogram
   • 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 = w + h and BD = w + h, so they match
   • AC = √(w² + h²) and BD = √(w² + h²), for every w and h
   • The diagonals bisect each other, so they are equal
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?

   • 13 m
   • 5 m
   • 19 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 5
   • Center (2, 4), radius 10
   • Center (4, 8), radius 5
   • Center (2, 4), radius 25
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 = 10 or x = -10
   • x = 5√2 or x = -5√2
   • x = 20 or x = -20
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?

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

   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
   • Acute: 26 is less than 18 + 20 = 38, so the angle at C is acute
   • Right, since 18 + 20 is close to 26
8. 8.

   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?

   • R is farthest at √41; P and Q tie at 5
   • 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

9. 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.

10. 9.

    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 = 3
    • x + y = 10
    • y = x − 1
    • x + y = 5
11. 10.

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

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