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IB Mathematics: Applications and Interpretation SL · 12.3: Coordinate Geometry · 12.3.1

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3D coordinates: distance and midpoint Problem set B

Show each midpoint and each distance. Use the calculator for the solving steps, and give 3 significant figures unless the question says exact.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Points A and B are \(A(-5, 3, 9)\) and \(B(7, -11, -1)\). M is the midpoint of AB.

    1. (a) Find the coordinates of M.
    2. (b) Find the distance from the origin to M. Give your answer to 3 significant figures.
    3. (c) P and Q are \(P(8, -4)\) and \(Q(-2, 5)\) in two dimensions. Find the midpoint of PQ.

  2. \(P(-3, 4, 1)\) and \(Q(2, -2, 8)\) are two points.

    1. (a) Find the distance PQ to 3 significant figures.
    2. (b) \(R(2, k, 3)\) is a distance 8 from \(S(-1, 5, -2)\). Find the possible values of k, to 3 significant figures.

  3. The midpoint of \(A(a, 3 b, -2)\) and \(B(2 b, a, 8)\) is \(M(\frac{11}{2}, 7, 3)\). Find the values of a and b.

  4. Find the equation of the line through \(P(-2, -5)\) and \(Q(4, 3)\). Give it in the form \(y = mx + c\) and in the form \(ax + by + d = 0\) with integer coefficients.

  5. A line L has equation \(2x + 3y - 12 = 0\). The point P is \((-1, 4)\).

    1. (a) Write down the gradient of L.
    2. (b) Find the equation of the line through P perpendicular to L.
    3. (c) Find the coordinates of the point where this line meets L.
    4. (d) Find the shortest distance from P to L.

In context

  1. A submarine travels in a straight line from \(A(-200, 350, -40)\) to \(B(160, -100, -180)\), where all coordinates are in metres and the z-coordinate is the depth relative to sea level (negative below it). The journey takes 2 minutes.

    1. (a) Find the distance travelled.
    2. (b) Find the average speed in metres per second.

  2. A greenhouse has a rectangular floor OABC and a horizontal ridge. The diagram shows the frame. O is the origin. The x-axis is along OA, the y-axis is along OC and the z-axis is vertical. All lengths are in metres. OA = 4, OC = 2.5. D and E are the ends of the ridge, 2.2 m above the ground, directly above the points 2 m along OA from O and C.

    OABCDE
    1. (a) Write down the coordinates of B and E.
    2. (b) M is the midpoint of the ridge DE. Find the coordinates of M.
    3. (c) A brace runs from M to a peg at \((2, 5.5, 0)\). Find the length of the brace.

  3. A packing crate is a cuboid 1.6 m long, 0.9 m wide and 0.7 m high. A straight rod rests with one end at corner A on the bottom and the other at the opposite corner G on the top.

    1.6 m0.9 m0.7 mABCDEFGH
    1. (a) Find the length of the diagonal AC of the base.
    2. (b) Find the length of the longest rod, the straight line AG.

  4. A cleaner wipes along the surface of a closed storage box 1.5 m long, 1 m wide and 0.6 m high, from corner A to the opposite corner G, in the shortest way. The cleaner stays on the surface. Find that distance.

    1.5 m1 m0.6 mABCDEFGH

Stretch

  1. ABCD is a square. A is \((3, -2)\) and C is \((-3, 4)\), and AC is a diagonal. B is the vertex with the larger x-coordinate.

    AC
    1. (a) Find the midpoint of AC.
    2. (b) Find the coordinates of B and D.
    3. (c) Find the area of the square.

  2. ABCD is a rhombus with \(A(-3, 2)\) and \(C(1, 6)\). The x-coordinate of B is 3.

    1. (a) Find the equation of the perpendicular bisector of AC.
    2. (b) The diagonal BD lies on this line. Find the coordinates of B and D.
    3. (c) Find the side length of the rhombus.

jmaths.xyz · AI SL · 12.3.1 3D coordinates: distance and midpoint