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 and BC = AD = 5, so both pairs of opposite sides are equal
   • AB = DC = √29, because one pair of equal sides is enough
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 = √(w² + h²) and BD = √(w² + h²), for every w and h
   • 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
3. 3.

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

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

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

   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
   • 20 km
   • 16 km
   • 8 km
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?

   • Obtuse, since AB² = 26 is bigger than both 18 and 20
   • Right, since the triangle is isosceles: 18 and 20 are close
   • Acute: 26 is less than 18 + 20 = 38, so the angle at C is acute
   • Right, since 18 + 20 is close to 26
7. 7.

   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?

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

   • 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

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?

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