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Math Coach Amy
Amy Ferguson Moncure

IB AA · 4B Three-Dimensional Coordinate Geometry

Speed, Diagonals, and Solids

Use distance and midpoint in a short situation.

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Key Ideas Easy Medium Hard

d = square root of x² + y² + z²

What to Remember

Distance from the origin

d = square root of x² + y² + z²

Average speed

Speed = distance ÷ time. Keep an exact root, then give 3 significant figures when the question asks for a decimal.

A square from one diagonal

The centre is the midpoint. Rotate the half-diagonal by 90° to find the other two vertices.

A path on the walls

Unfold the room into a net. The straight line on the net is the shortest crawl.

Easy

Easy

A distance from the origin, or across a floor.

Example
Find the exact distance from (6, 8) to the origin.

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 0)² + (8 − 0)²

d = square root of 36 + 64

d = square root of 100

d = 10

Answer: 10

1
Find the exact distance from (3, 4, 12) to the origin.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 0)² + (4 − 0)² + (12 − 0)²

d = square root of 9 + 16 + 144

d = square root of 169

d = 13

Answer: 13

2
Find the exact distance from (2, 3, 6) to the origin.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − 0)² + (3 − 0)² + (6 − 0)²

d = square root of 4 + 9 + 36

d = square root of 49

d = 7

Answer: 7

3
Find the exact distance from (1, 4, 8) to the origin.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (1 − 0)² + (4 − 0)² + (8 − 0)²

d = square root of 1 + 16 + 64

d = square root of 81

d = 9

Answer: 9

4
Find the exact distance from (6, 8, 0) to the origin.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (6 − 0)² + (8 − 0)² + (0 − 0)²

d = square root of 36 + 64 + 0

d = square root of 100

d = 10

Answer: 10

5
A hallway floor is a rectangle 6 m by 8 m. How far is it from one corner of the floor to the opposite corner?
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)²

d = square root of (6 − 0)² + (8 − 0)²

d = square root of 36 + 64

d = square root of 100

d = 10

Answer: 10 m

Medium

Medium

A flight, a tent, and a midpoint.

Example
A bird leaves a perch at (0, 0, 2) and flies in a straight line to a branch at (8, 6, 5). The flight takes 4 seconds. Distances are in metres. Find the average speed to 3 significant figures.

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (8 − 0)² + (6 − 0)² + (5 − 2)²

d = square root of 64 + 36 + 9

d = square root of 109

Speed = square root of 109 ÷ 4

Speed is about 2.61 m/s

Answer: about 2.61 m/s

1
A tent has floor corners A(0, 0, 0), B(8, 0, 0), C(8, 6, 0), and D(0, 6, 0). The highest point is F(4, 3, 5). All lengths are in metres. State the height of the tent.
Show Answer

Height is the z-coordinate of the highest point.

The height is 5 m.

Answer: 5 m

2
Using the same floor, an extra pole runs from the midpoint of BC to E(2, 3, 4). Find the exact length of the pole.
Show Answer

The midpoint of B(8, 0, 0) and C(8, 6, 0) is (8, 3, 0).

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − 8)² + (3 − 3)² + (4 − 0)²

d = square root of 36 + 0 + 16

d = square root of 52

d = 2square root of 13

Answer: 2square root of 13 m

3
Find the midpoint of (1, −5, 3) and (5, 1, 7), then find its distance from the origin.
Show Answer

M = (x₁ + x₂2, y₁ + y₂2, z₁ + z₂2)

M = (1 + 52, −5 + 12, 3 + 72)

M = (62, −42, 102)

M = (3, −2, 5)

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 0)² + (−2 − 0)² + (5 − 0)²

d = square root of 9 + 4 + 25

d = square root of 38

Answer: midpoint (3, −2, 5), distance square root of 38

4
Find the exact distance from (9, 12, 8) to the origin.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (9 − 0)² + (12 − 0)² + (8 − 0)²

d = square root of 81 + 144 + 64

d = square root of 289

d = 17

Answer: 17

5
A lift moves from (1, 2, 0) to (5, 5, 12), in metres, in 5 seconds. Find the exact distance.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (5 − 1)² + (5 − 2)² + (12 − 0)²

d = square root of 16 + 9 + 144

d = square root of 169

d = 13

Answer: 13

Hard

Hard

A square, a rhombus, a room, and a right angle.

Example
A(1, 2) and C(7, 4) are opposite vertices of square ABCD. Find B and D.

The midpoint of AC is the centre.

M = (4, 3)

Half of AC is (3, 1).

Rotate (3, 1) by 90° to get (−1, 3).

B = (4, 3) + (−1, 3) = (3, 6)

D = (4, 3) − (−1, 3) = (5, 0)

Each side has length 2square root of 5.

Answer: B(3, 6) and D(5, 0), in either order

1
A(1, 1) and C(5, 3) are opposite ends of a diagonal of rhombus ABCD. The gradient of AB is 3. Find the equation of BD, then find B and D, and the side length.
Show Answer

The diagonals of a rhombus bisect each other at right angles.

Midpoint of AC: (3, 2)

Gradient of AC is 12, so the gradient of BD is −2.

y − 2 = −2(x − 3)

y = −2x + 8

Let B be (3 + t, 2 − 2t), so it lies on that line.

Gradient of AB is 3:

(2 − 2t) − 1(3 + t) − 1 = 3

1 − 2t2 + t = 3

1 − 2t = 6 + 3t

t = −1

B = (2, 4) and D = (4, 0)

Each side has length square root of 10.

Answer: BD: y = −2x + 8; B(2, 4), D(4, 0); side square root of 10

2
A room is 2 m by 6 m by 3 m. A fly goes in a straight line to the furthest corner. A spider then crawls along the walls to that same corner. Find the fly's distance, and the shortest distance the spider might crawl.
Show Answer

Flying through the room uses the space diagonal.

d = square root of 2² + 6² + 3²

d = square root of 4 + 36 + 9

d = square root of 49 = 7

The shortest crawl unfolds two faces into one rectangle.

The three possible straight crawls are square root of 61, square root of 73, and square root of 85.

The shortest of those is square root of 61.

Flying is shorter than crawling along the walls.

Answer: fly 7 m, spider square root of 61 m

3
Using the same floor, an extra pole runs from the midpoint of BC to E(2, 3, 4). Find the exact length of the pole.
Show Answer

The midpoint of B(8, 0, 0) and C(8, 6, 0) is (8, 3, 0).

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − 8)² + (3 − 3)² + (4 − 0)²

d = square root of 36 + 0 + 16

d = square root of 52

d = 2square root of 13

Answer: 2square root of 13 m

4
A bird leaves a perch at (0, 0, 2) and flies in a straight line to a branch at (8, 6, 5). The flight takes 4 seconds. Distances are in metres. Find the average speed to 3 significant figures.
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (8 − 0)² + (6 − 0)² + (5 − 2)²

d = square root of 64 + 36 + 9

d = square root of 109

Speed = square root of 109 ÷ 4

Speed is about 2.61 m/s

Answer: about 2.61 m/s

5
A(1, 2, 0), B(4, 6, 0), and C(4, 6, 5) are the vertices of a triangle. Show that the triangle is right-angled, and give the exact length of the hypotenuse.
Show Answer

Side AB.

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − 1)² + (6 − 2)² + (0 − 0)²

d = square root of 9 + 16 + 0

d = square root of 25

d = 5

Side BC.

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − 4)² + (6 − 6)² + (5 − 0)²

d = square root of 0 + 0 + 25

d = square root of 25

d = 5

Side AC.

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − 1)² + (6 − 2)² + (5 − 0)²

d = square root of 9 + 16 + 25

d = square root of 50

d = 5square root of 2

AB = 5 and BC = 5, so AB² + BC² = 25 + 25 = 50.

AC = 5square root of 2, and 5² × 2 = 50.

The squares match, so the angle at B is a right angle.

The hypotenuse is AC = 5square root of 2.

Answer: right-angled at B, hypotenuse 5square root of 2

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