3D coordinates: distance and midpoint Problem set A Worked solutions

Points A and B are \(A(4, -6, 10)\) and \(B(-2, 8, 2)\). M is the midpoint of AB.

1. (a) Find the coordinates of M.

   • Add each pair of coordinates and halve
   • \(M = (1, 1, 6)\)

2. (b) Find the distance from the origin to M. Give your answer to 3 significant figures.

   • \(OM = \sqrt{1+1+36} = 6.16\)

3. (c) P and Q are \(P(-3, 7)\) and \(Q(9, -2)\) in two dimensions. Find the midpoint of PQ.

   • \(\left(\frac{-3+(9)}{2}, \frac{7+(-2)}{2}\right) = (3, 2.5)\)

\(P(2, -1, 5)\) and \(Q(7, 3, -4)\) are two points.

1. (a) Find the distance PQ to 3 significant figures.

   • \(PQ = \sqrt{(7-(2))^2+(3-(-1))^2+(-4-(5))^2}\)
   • \(= \sqrt{122} = 11.0\)

2. (b) \(R(k, 2, -1)\) is a distance 9 from \(S(4, -3, 5)\). Find the possible values of k, to 3 significant figures.

   • \(RS^2 = 81\): \((k-(4))^2 + 61 = 81\)
   • \((k-(4))^2 = 20\)
   • \(k = −0.472\) or \(k = 8.47\) (two values: the point can be on either side of S)

The midpoint of \(A(a, 2 b, 3)\) and \(B(3 b, a, 5)\) is \(M(4, 5, 4)\). Find the values of a and b.

• Each coordinate of M is the average: \(\frac{a+3 b}{2} = 4\) and \(\frac{2 b+a}{2} = 5\)
• \(a + 3 b = 8\) and \(a + 2 b = 10\)
• Solve together: \(a = 14\), \(b = -2\)
• Check with the third coordinate: \(\frac{3+5}{2} = 4\)

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

• Gradient \(= \frac{-4-(2)}{5-(-3)} = - \frac{3}{4}\)
• \(y - (2) = - \frac{3}{4}(x - (-3))\)
• \(y = - \frac{3}{4}x - \frac{1}{4}\)
• Multiply by 4 and collect terms: \(3x + 4y + 1 = 0\)

A line L has equation \(3x - 4y + 12 = 0\). The point P is \((7, 1)\).

1. (a) Write down the gradient of L.

   • Rearrange: \(y = \frac{3}{4}x + 3\)
   • Gradient \(\frac{3}{4}\)

2. (b) Find the equation of the line through P perpendicular to L.

   • Perpendicular gradient \(= -1 \div \frac{3}{4} = - \frac{4}{3}\)
   • \(y = - \frac{4}{3}x + \frac{31}{3}\)

3. (c) Find the coordinates of the point where this line meets L.

   • Solve the two equations together on the calculator
   • Meeting point \((\frac{88}{25}, \frac{141}{25})\)

4. (d) Find the shortest distance from P to L.

   • \(\sqrt{(7-\frac{88}{25})^2+(1-\frac{141}{25})^2} = 5.80\)

A drone flies in a straight line from \(A(10, 25, 0)\) to \(B(130, 95, 48)\), where all coordinates are in metres and the z-coordinate is the height above the ground. The flight takes 18 seconds.

1. (a) Find the distance travelled.

   • \(\sqrt{(130-(10))^2+(95-(25))^2+(48-(0))^2}\)
   • \(= \sqrt{21604} = 147\) m

2. (b) Find the average speed in metres per second.

   • Speed = distance ÷ time = \(147 \div 18\)
   • \(8.17\) m/s

A tent 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 = 2.4, OC = 3.6. D and E are the ends of the ridge, 1.5 m above the ground, directly above the points 1.2 m along OA from O and C.

1. (a) Write down the coordinates of B and E.

   • B is above nothing: on the ground, \((2.4, 3.6, 0)\)
   • E is at the same height as D and the same x: \((1.2, 3.6, 1.5)\)

2. (b) M is the midpoint of the ridge DE. Find the coordinates of M.

   • \(D = (1.2, 0, 1.5)\), \(E = (1.2, 3.6, 1.5)\)
   • \(M = (1.2, 1.8, 1.5)\)

3. (c) A rope runs from M to a peg at \((1.2, 6, 0)\). Find the length of the rope.

   • \(\sqrt{(1.2-1.2)^2+(6-1.8)^2+(0-1.5)^2} = 4.46\) m

A hall is a cuboid 7.5 m long, 5.2 m wide and 2.9 m high. A string is pulled tight from corner A on the floor to the opposite corner G on the ceiling.

1. (a) Find the length of the diagonal AC of the base.

   • \(AC^2 = 7.5^2 + 5.2^2 = 83.29\)
   • \(AC = 9.13\) m

2. (b) Find the length of the string, the straight line AG.

   • \(AG^2 = AC^2 + 2.9^2 = 91.70\)
   • \(AG = 9.58\) m

A spider walks over the surface of a closed box 12 cm long, 5 cm wide and 4 cm high, from corner A to the opposite corner G. Find the shortest distance it can walk. The spider stays on the surface.

• Unfold two faces into a rectangle. Three different rectangles are possible.
• \(\sqrt{(5+4)^2+12^2} = 15.0\); \(\sqrt{(12+4)^2+5^2} = 16.8\); \(\sqrt{(12+5)^2+4^2} = 17.5\)
• The shortest is 15.0 cm

ABCD is a square. A is \((-1, 2)\) and C is \((5, -2)\), and AC is a diagonal. B is the vertex with the larger x-coordinate.

1. (a) Find the midpoint of AC.

   • \(M = \left(\frac{-1+(5)}{2}, \frac{2+(-2)}{2}\right) = (2, 0)\)

2. (b) Find the coordinates of B and D.

   • M to C: \((3, -2)\). Turn through 90°: \((2, 3)\)
   • \(B = M + (2, 3) = (4, 3)\)
   • \(D = M - (2, 3) = (0, -3)\)

3. (c) Find the area of the square.

   • \(AB^2 = (4-(-1))^2+(3-(2))^2 = 26.0\)
   • Area \(= AB^2 = 26.0\)

ABCD is a rhombus with \(A(-1, 0)\) and \(C(5, 4)\). The x-coordinate of B is 4.

1. (a) Find the equation of the perpendicular bisector of AC.

   • Midpoint of AC: \(M = (2, 2)\)
   • Gradient of AC \(= \frac{2}{3}\), so the perpendicular gradient is \(- \frac{3}{2}\)
   • \(y = - \frac{3}{2}x + 5\)

2. (b) The diagonal BD lies on this line. Find the coordinates of B and D.

   • At \(x = 4\): \(y = -1\), so \(B = (4, -1)\)
   • The diagonals bisect each other, so \(D = 2M - B = (0, 5)\)

3. (c) Find the side length of the rhombus.

   • \(AB = \sqrt{(4-(-1))^2+(-1-(0))^2} = 5.10\)