Geometry & trig 1 · Distance & midpoint in 3D

Module 19 of 47Lesson 1 of 8 · Geometry & trig 1 · Distance & midpoint in 3D

3D geometry & trigonometry · Essential understanding 1

Distance & midpoint in 3D

IB AI SL 3.1with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: A point in space needs three numbers: (x, y, z) — along, across and up. The distance between two points is Pythagoras twice, and the midpoint is the average of each coordinate.

By the end of this lesson you can:

read and plot a point (x, y, z) in 3D
find the distance between two points in 3D
find the midpoint of two points in 3D
use these to solve problems about rooms, boxes and flight paths

🧑🏾 Imagine this…

Stand in the corner of your classroom floor. Call that corner the origin (0, 0, 0). Walk 3 m along one wall (x), 2 m along the other wall (y), then climb a ladder 2.5 m up (z). You are now at the point (3, 2, 2.5). Three numbers pin down exactly where you are. A fly in the opposite top corner is a point too — and the straight line between you and the fly is a distance in 3D.

Three axes meet at the origin. x goes along, y goes across (drawn slanting backwards), z goes up.

Coordinates (x, y, z)

three numbers that give a point's position: x along, y across, z up (the height).

Origin

the point (0, 0, 0) where the three axes meet.

Space diagonal

the straight line from one corner of a box to the opposite corner, passing through the inside of the box.

Midpoint

the point exactly halfway between two points.

Use Pythagoras on the floor first (dashed), then again going up. That gives the distance d.

★ Distance and midpoint (both in the formula booklet)

Distance between $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ :
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
Midpoint of AB: $M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2},\ \dfrac{z_1 + z_2}{2}\right)$
Why it works: the floor diagonal comes from Pythagoras on the floor; then you go up and use Pythagoras a second time.

⚠️ Negatives and squares

Subtract carefully: from (2, 1, 4) to (5, −3, 16) the y change is −3 − 1 = −4, and (−4)² = +16. It does not matter which point you call 1 and which 2 — squaring removes the sign — but you must subtract the same point in all three brackets.

⚠️ Do not forget the root

The sum under the square root is d², not d. A very common slip is to stop at 169 instead of √169 = 13. Always finish with the square root.

💡 Command terms

Find the distance: give a number with units. Show that AB = 13: write the working fully and end on exactly 13. Write down a midpoint: no working needed, just average each coordinate.

🧪 Try it yourselfDistance or midpoint?

Quick checks using numbers from this lesson. Think first, then pick.

Worked examples

Example 1.1Distance between two points

Find the distance between P(2, 1, 4) and Q(5, −3, 16).

🧠 Think it through

Find the three differences: x: 5 − 2, y: −3 − 1, z: 16 − 4.
Square each, add, then take the square root.

✍️ By hand

d = \sqrt{(5 - 2)^2 + (-3 - 1)^2 + (16 - 4)^2}

= \sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169}

d = 13

🔢 On your TI-Nspire CX II

On a Calculator page press ctrl💻 Ctrl then the square-root key to get √(, and type the whole sum inside.
Use the (−) key for negative numbers, not the minus key.
Check the brackets: every difference is in its own bracket, then squared.

Exactly 13 — no rounding needed

✅ Answer PQ = 13 units.

Example 1.2Midpoint and distance together

A is the point (−2, 5, 7) and B is the point (6, −1, −3).

(a) Find the midpoint M of AB.
(b) Find the length of AB.

🧠 Think it through

Midpoint: add the two x values and halve; then the y values; then the z values.
Distance: three differences, square, add, root. Round to 3 s.f. at the end.

✍️ By hand

M = \left(\dfrac{-2 + 6}{2},\ \dfrac{5 + (-1)}{2},\ \dfrac{7 + (-3)}{2}\right) = (2, 2, 2)

AB = \sqrt{(6 - (-2))^2 + (-1 - 5)^2 + (-3 - 7)^2} = \sqrt{64 + 36 + 100} = \sqrt{200}

AB = 14.142\ldots = 14.1

(3 s.f.)

🔢 On your TI-Nspire CX II

Midpoint: type each average, for example (−2+6)/2, or do these three in your head.
Distance: type the whole sum inside √( ) and press enter ↵💻 Enter. The TI gives a decimal.

Midpoint (2, 2, 2); distance 14.1421 → 14.1

✅ Answer (a) M(2, 2, 2). (b) AB = √200 = 14.1 (3 s.f.).

Because 200 = 100 × 2, the exact answer is also 10√2. The 3 s.f. value is the safe one to write.

Example 1.3A drone near the tower

The control box of a drone is on the ground in Gaborone. Taking the control box as the origin, distances are in metres: x is east, y is north, z is up. The drone's camera is at the point D(120, −80, 45). The pilot's eyes are 2 m above the control box, at E(0, 0, 2). How far is the drone from the pilot's eyes?

🧠 Think it through

Two points in 3D → use the distance formula.
The differences are 120 − 0, −80 − 0 and 45 − 2 (careful: z is 43, not 45).
Round to 3 s.f. and give the unit.

✍️ By hand

ED = \sqrt{120^2 + (-80)^2 + 43^2} = \sqrt{14400 + 6400 + 1849} = \sqrt{22649}

ED = 150.49\ldots = 150

m (3 s.f.)

🔢 On your TI-Nspire CX II

Type √(120^2+80^2+43^2) and press enter ↵💻 Enter. (Squaring a negative gives the same as squaring the positive, so 80 is fine here.)
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

150.496 → 150 m

✅ Answer About 150 m.

Check the answer is sensible: it must be longer than the biggest single difference (120).

Example 1.4The longest rod in a room

A storeroom is a cuboid 6 m long, 4 m wide and 3 m high. A builder wants to store a straight pipe. Find the longest pipe that fits, and the midpoint of its path if one end is at a floor corner O(0, 0, 0).

🧠 Think it through

Put the origin at a floor corner. The opposite top corner is G(6, 4, 3).
The longest straight pipe goes corner-to-corner: the space diagonal OG.
Midpoint: halve each coordinate of G, since O is (0, 0, 0).

✍️ By hand

OG = \sqrt{6^2 + 4^2 + 3^2} = \sqrt{36 + 16 + 9} = \sqrt{61}

OG = 7.81\ldots = 7.81

m (3 s.f.)

M = \left(\dfrac{6}{2},\ \dfrac{4}{2},\ \dfrac{3}{2}\right) = (3, 2, 1.5)

🔢 On your TI-Nspire CX II

Type √(6^2+4^2+3^2) enter ↵💻 Enter.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

7.81025 → 7.81 m

✅ Answer Longest pipe 7.81 m; its midpoint is (3, 2, 1.5).

Shortcut for any cuboid: space diagonal = √(l² + w² + h²).

Example 1.5Finding an unknown coordinate

The points A(1, 2, 3) and B(k, −2, 0) are 7 units apart, and k is positive.

(a) Show that (k − 1)² = 24.
(b) Find k, correct to 3 significant figures.
(c) Find the midpoint of AB.

🧠 Think it through

Write distance² = 7² = 49 using the formula, so there is no root to deal with.
Solve (k − 1)² = 24: take ± the root, then keep the positive k.
Use the unrounded k in the midpoint.

✍️ By hand

AB^2 = (k - 1)^2 + (-2 - 2)^2 + (0 - 3)^2 = (k - 1)^2 + 16 + 9

7^2 = (k - 1)^2 + 25 \Rightarrow (k - 1)^2 = 24

k - 1 = \pm\sqrt{24} = \pm 4.899

k = 5.90

-3.90

. Since

k > 0

k = 5.90

M = \left(\dfrac{1 + 5.899}{2},\ \dfrac{2 + (-2)}{2},\ \dfrac{3 + 0}{2}\right) = (3.45,\ 0,\ 1.5)

🔢 On your TI-Nspire CX II

Use the numerical solver: type nSolve((k−1)^2=24,k) on a Calculator page and press enter ↵💻 Enter.
📍 Where to find it on the TI-Nspire
nSolve( is in menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve
The solver finds one answer. To make it find the positive one, add the condition: nSolve((k−1)^2=24,k)|k>0 (the | key is the “with” key).
📍 Where to find it on the TI-Nspire
nSolve( is in menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve
⌨️ the “such that” bar |: type | (Shift + \ on the keyboard)
Then type the midpoint x: (1+5.89898)/2.

k = 5.89898 → 5.90; midpoint x = 3.45

✅ Answer (a) shown above. (b) k = 5.90. (c) Midpoint (3.45, 0, 1.5).

If a question says “k is positive” (or gives a range) it is telling you to throw away one of the two roots.

Challenge questions

⭐ Challenge 1.6A triangle in spacePaper 1 style

The points A(1, 0, 2), B(4, 4, 2) and C(−3, 3, 2) are the corners of a triangle.

(a) Find the length of AB.
(b) Find the length of AC.
(c) Hence state what type of triangle ABC is, giving a reason.
(d) Find the midpoint of BC.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Use the distance formula twice. Notice that every z is 2 — the triangle is flat, parallel to the floor.
Equal sides → isosceles.
Midpoint of BC: average the coordinates of B and C.

✍️ By hand

AB = \sqrt{3^2 + 4^2 + 0^2} = 5

;

AC = \sqrt{(-4)^2 + 3^2 + 0^2} = 5

AB = AC

, so the triangle is isosceles.

M = \left(\dfrac{4 + (-3)}{2},\ \dfrac{4 + 3}{2},\ \dfrac{2 + 2}{2}\right) = (0.5,\ 3.5,\ 2)

🔢 On your TI-Nspire CX II

Type √(3^2+4^2) and √(4^2+3^2) — both give 5.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

✅ Answer (a) 5. (b) 5. (c) Isosceles, because AB = AC. (d) (0.5, 3.5, 2).

Extra: BC² = 7² + 1² = 50 = 5² + 5², so the triangle is also right-angled at A (Pythagoras). Can you see why?

⭐ Challenge 1.7The zip-line at KasanePaper 2 style

A zip-line near Kasane runs from a top platform T to a landing platform L. Using a point on the ground below T as origin, the coordinates are T(0, 0, 30) and L(80, 45, 2), all in metres.

(a) Find the length of the zip-line, to 3 s.f.
(b) Find the coordinates of the midpoint M of TL.
(c) A rider takes 8.5 seconds to travel from T to L. Find the average speed in m s⁻¹ and in km h⁻¹.
(d) A safety post stands on the ground directly below M. How tall must it be to touch the line at M?

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Differences: 80, 45 and 2 − 30 = −28.
Midpoint: halve the sum of each coordinate.
$\text{Speed} = \dfrac{\text{distance}}{\text{time}}$ ; multiply m s⁻¹ by 3.6 for km h⁻¹.
The post reaches up to the z-coordinate of M.

✍️ By hand

TL = \sqrt{80^2 + 45^2 + (-28)^2} = \sqrt{9209} = 95.96\ldots = 96.0

M = \left(40,\ 22.5,\ 16\right)

\text{speed} = \dfrac{95.96}{8.5} = 11.3

m s⁻¹,

11.29 \times 3.6 = 40.6

km h⁻¹

Post height = z-coordinate of M = 16 m.

🔢 On your TI-Nspire CX II

Type √(80^2+45^2+28^2) enter ↵💻 Enter.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(
Use the last answer (ans) for the next lines: type ans/8.5 then ans*3.6.

96.0 m; 11.3 m/s; 40.6 km/h

✅ Answer (a) 96.0 m. (b) M(40, 22.5, 16). (c) 11.3 m s⁻¹ = 40.6 km h⁻¹. (d) 16 m.

Next: Lesson 2 · Volume & surface area of solids

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL Geometry & trig 1 · Distance & midpoint in 3D

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Distance & midpoint in 3D

Worked examples

Challenge questions