Mathematics • Year 10 • Unit 2 • Lesson 15
Distance and Midpoint, Skill Drill
Build fluency with the two coordinate-geometry tools from Lesson 15: the distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) and the midpoint formula M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Then use them backwards to find an unknown endpoint.
1. I do, fully worked example
Distance between two points using the distance formula. Read each reason.
Problem. Find the distance between A(3, −2) and B(7, 4).
Step 1, Label the coordinates.
(x₁, y₁) = (3, −2) (x₂, y₂) = (7, 4)
Reason: labelling avoids subtracting the wrong way around.
Step 2, Compute the run and the rise (use brackets around negatives).
x₂ − x₁ = 7 − 3 = 4
y₂ − y₁ = 4 − (−2) = 4 + 2 = 6
Step 3, Square, add, and take the root.
d = √(4² + 6²) = √(16 + 36) = √52
Step 4, Simplify the surd.
√52 = √(4 × 13) = 2√13 ≈ 7.21 units
Answer: d = √52 = 2√13 ≈ 7.21 units.
2. We do, fill in the missing steps
Midpoint of an interval. Fill in each blank. 5 marks
Problem. Find the midpoint M of the interval joining P(−4, 6) and Q(8, −2).
Step 1, Label: (x₁, y₁) = ( ____, ____ ), (x₂, y₂) = ( ____, ____ ).
Step 2, Compute the average x:
(−4 + 8) / 2 = ____ / 2 = ____
Step 3, Compute the average y:
(6 + (−2)) / 2 = ____ / 2 = ____
Step 4, Write M as an ordered pair:
M = ( ____, ____ )
Step 5, Quick check: Is M halfway between P and Q? Distance PM should equal distance MQ. (You don't need to compute both, just sanity-check directions.)
3. You do, independent practice
Give exact answers using surds where appropriate. Always check that your midpoint is between the two points.
Foundation, straight-line distances and midpoints
3.1 Find the distance between (0, 0) and (3, 4). 1 mark
3.2 Find the distance between (1, 2) and (4, 6). 1 mark
3.3 Find the midpoint of (2, 6) and (8, −4). 1 mark
3.4 Find the midpoint of (−3, 5) and (7, −1). 1 mark
Standard, negatives, exact surds
3.5 Find the exact distance between (1, −2) and (4, 2). 2 marks
3.6 Find the exact distance between (−5, 1) and (3, 7), and simplify the surd. 2 marks
Extension, find the endpoint, geometry checks
3.7 M(4, −3) is the midpoint of AB. If A is (1, 2), find B. Show the formula. 3 marks
3.8 Triangle PQR has vertices P(1, 1), Q(5, 1) and R(3, 5). (a) Find the length of each side. (b) Is the triangle isosceles? Justify. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (P(−4, 6), Q(8, −2))
Step 1: (x₁, y₁) = ( −4, 6 ); (x₂, y₂) = ( 8, −2 ). Step 2: average x = 4/2 = 2. Step 3: average y = 4/2 = 2. Step 4: M = ( 2, 2 ). Step 5: ✓.
3.1, (0,0) to (3,4)
d = √(3² + 4²) = √(9 + 16) = √25 = 5.
3.2, (1,2) to (4,6)
d = √(3² + 4²) = √25 = 5.
3.3, midpoint (2,6) and (8,−4)
M = ((2+8)/2, (6+(−4))/2) = (10/2, 2/2) = (5, 1).
3.4, midpoint (−3,5) and (7,−1)
M = ((−3+7)/2, (5+(−1))/2) = (2, 2). (2, 2).
3.5, (1,−2) to (4,2)
Δx = 3, Δy = 4. d = √(9 + 16) = √25 = 5 units (exact).
3.6, (−5,1) to (3,7)
Δx = 8, Δy = 6. d = √(64 + 36) = √100 = 10 units.
3.7, Find B given M(4,−3) and A(1,2)
x_B = 2(4) − 1 = 7; y_B = 2(−3) − 2 = −8. B = (7, −8). Check midpoint of (1,2) and (7,−8): ((1+7)/2, (2−8)/2) = (4, −3) ✓.
3.8, Triangle PQR
PQ = √((5−1)² + 0²) = 4. PR = √((3−1)² + (5−1)²) = √(4 + 16) = √20 = 2√5. QR = √((3−5)² + (5−1)²) = √(4 + 16) = √20 = 2√5. (b) PR = QR = 2√5 ≠ PQ, so the triangle is isosceles (two equal sides PR and QR).