Proving Geometry Facts with Vectors · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Proving lines parallel with vectors

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. In triangle OAB, OA = a and OB = b (both measured from O). Write the vector AB in terms of a and b.

    • b − a
    • a − b
    • a + b
    • −a − b

  2. In triangle OAB, OA is a and OB is b, both from O. Write vector AB in terms of a and b.

    • a - b
    • b - a
    • a + b

  3. OA is a and OB is b. M is the midpoint of AB. Write OM in terms of a and b.

    • Half of (a + b)
    • a + b
    • Half of (a - b)

  4. PQ is 2a plus 6b and RS is a plus 3b. The lines PQ and RS are parallel. True or false?

    Circle one:   True   False

  5. Vector u = 6a + 9b is parallel to vector v = 2a + kb. Find the value of k.

    Answer: ______________

  6. AB = 4a − 2b and CD = 6a − 3b. What number k makes CD = k × AB?

    • 1.5
    • 2
    • 0.5
    • 3

  7. OA = a and OB = b. M is the midpoint of AB. Write OM in terms of a and b.

    • ½(a + b)
    • a + b
    • ½(a − b)
    • ½a + b

  8. You found PQ = 2a + b and QR = 2a + b, and P, Q, R all connect through the shared point Q. True or false: this proves P, Q and R lie on the same straight line.

    Circle one:   True   False

  9. PQ and QR both equal 2a plus b and share point Q. So P, Q and R lie on one straight line. True or false?

    Circle one:   True   False

  10. AB = ¾a − ½b and CD = 3a − 2b. Find the number k such that CD = k × AB.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Proving lines parallel with vectors W1-mt_A00MA7Ecgf-s1

  1. b − a · To travel from A to B, undo the trip to A first, then do the trip to B.
  2. b - a · Undo the trip to A first, then do the trip to B.
  3. Half of (a + b) · Get to A, then go halfway along AB.
  4. True · PQ is exactly 2 times RS, so they point the same way.
  5. 3 · Being parallel means u is some plain number times v, so match up the a-parts and b-parts.
  6. 1.5 · Match up the a-parts and the b-parts to solve for k, then check both agree.
  7. ½(a + b) · Build the route to M in stages: get to A, then go halfway along AB.
  8. True · Equal vectors point the same direction, and sharing point Q locks them onto one line.
  9. True · Equal vectors share a direction and Q locks them onto one line.
  10. 4 · Even with fraction coefficients, matching the a-parts still pins down k.
Worksheet · LightMySky