The distance formula

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   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?

   • AC = √(w² + h²) and BD = √(w² + h²), for every w and h
   • The diagonals bisect each other, so they are equal
   • 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
2. 2.

   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
3. 3.

   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 = 20 or x = -20
   • x = 10 or x = -10
   • x = 2√5 or x = -2√5
4. 4.

   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
5. 5.

   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
   • Acute: 26 is less than 18 + 20 = 38, so the angle at C is acute
   • Right, since the triangle is isosceles: 18 and 20 are close
   • Right, since 18 + 20 is close to 26
6. 6.

   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
   • (0, 0) and (6, 0)
   • (-2, 0) and (8, 0)
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 is farthest, since it is listed first
   • R is farthest at √41; P and Q tie at 5
   • P, Q and R are each 5 away from the base

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?

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