Year 10 Extended · Unit 2 Generalizing, coordinate geometry and linear programming · Lesson 3 of 6

What do the middles of any four-sided shape always make?

A drawing of one quadrilateral shows a property of that one. Putting letters in the coordinates turns midpoints, lengths and gradients into a proof about every quadrilateral at once.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: find the midpoint of (−2, 5) and (6, 1).
    Answer(2, 3). Add the coordinates and halve each sum.
  2. From Year 9: are lines with gradients 2 and −1/2 parallel, perpendicular or neither?
    AnswerPerpendicular, because 2 × (−1/2) = −1.
  3. From lesson 2: why do two different odd numbers need two different letters?
    AnswerOne letter would make them equal.
Words used in this lesson
midpoint
the point halfway between two points. For example: the midpoint of (0, 0) and (10, 6) is (5, 3).
gradient
how much a line goes up for each step across. For example: a line through (0, 1) and (3, 7) has gradient 2.
diagonal
a line joining two corners that are not neighbours. For example: AC in the quadrilateral ABCD.
parallel
never meeting, with the same gradient. For example: y = 2x + 1 and y = 2x − 4.
perpendicular
at a right angle; gradients multiply to −1. For example: gradients 3 and −1/3.
rhombus
a quadrilateral with four equal sides. For example: a diamond shape.

The middles of the sides

A tiler in Montevideo cuts a four-sided slab, then marks the middle of each side and joins the four marks in order to make an inner frame. She has tried a handful of slabs and wonders whether the frame always has a pattern, or whether it changes from slab to slab.

Predict first: what do you think the inner shape always is? What could change it? Write a prediction and a reason, then move C and D in the figure. A and B stay where they are.

3
4
−3
5
move C or Dthe midpoint quadrilateral
not yet showngradient of M1M2 and M4M3

Move C or D with the sliders. The figure will then name the shape the midpoints make and give the gradients of its sides.

Show me the steps
  1. C = (3, 4), D = (−3, 5): the diagonals are neither equal nor perpendicular, and the midpoint quadrilateral is only a parallelogram.
  2. C = (3, 3), D = (−4, 1): AC² = 106 and BD² = 106. The diagonals are equal, so the midpoint quadrilateral is a rhombus.
  3. C = (1, 5), D = (−4, 5): AC and BD are perpendicular (gradients 1 and −1). The midpoint quadrilateral is a rectangle.
  4. C = (2, 4), D = (−1, 4): AC² = 100 = BD² and the diagonals are perpendicular, so the midpoint quadrilateral is a square.

Letters in the coordinates make the shape general. A proof about every quadrilateral can place one vertex at the origin and one side along the x-axis, because where the axes go does not change the shape. Write the other vertices with letters, and use 2a, 2b, … so that midpoints have no fractions.

Then use three tools: the midpoint ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2), the squared distance (x₂ − x₁)² + (y₂ − y₁)², and the gradient. Equal gradients mean parallel; a product of −1 means perpendicular. End with a sentence that names the property.

Two common mistakes. Choosing coordinates that quietly make the shape special: putting A at (0, 0), B at (a, 0) and D at (0, d) gives every quadrilateral a right angle at A, so the proof covers only quadrilaterals with a right angle, not every quadrilateral. And testing perpendicular with m₁ = −m₂: gradients 2 and −2 are not perpendicular, because 2 × (−2) = −4. The product has to be −1.

Worked examples

1. A(1, 1), B(7, 3), C(9, 9), D(1, 7). Show that the midpoints of the sides make a parallelogram.

  1. The midpoints are P(4, 2), Q(8, 6), R(5, 8), S(1, 4).
  2. PQ: gradient 4 ÷ 4 = 1 and SR: (8 − 4) ÷ (5 − 1) = 1. QR: (8 − 6) ÷ (5 − 8) = −2/3 and PS: (4 − 2) ÷ (1 − 4) = −2/3. Both pairs of opposite sides are parallel, so PQRS is a parallelogram.

2. Prove that the diagonals of a rectangle are equal.

  1. Place it with A(0, 0), B(w, 0), C(w, h), D(0, h). Then AC² = w² + h² and BD² = (0 − w)² + (h − 0)² = w² + h².
  2. AC² = BD², so AC = BD for every width w and height h: the diagonals are equal.

3. Prove that the diagonals of a rhombus are perpendicular.

  1. Place it with O(0, 0), P(p, 0), Q(p + q, r), R(q, r). The sides are equal, so OR² = OP²: q² + r² = p², which gives r² = p² − q².
  2. OQ has gradient r ÷ (p + q) and PR has gradient r ÷ (q − p). Their product is r² ÷ (q² − p²) = r² ÷ (−r²) = −1, so the diagonals are perpendicular.

Your go. The first step is given. A right-angled triangle has O(0, 0), P(2a, 0) and Q(0, 2b). Prove that the midpoint M of the hypotenuse PQ is the same distance from O, P and Q.

  1. The midpoint of PQ is M(a, b), and MO² = a² + b².
  2. Now work out MP² and MQ², and finish with a sentence.
Check the answer

MP² = (2a − a)² + (0 − b)² = a² + b² and MQ² = (0 − a)² + (2b − b)² = a² + b². All three squared distances are a² + b², so M is the same distance from O, P and Q, for every a and b.

Check before you go on

1. In a proof, A is (0, 0), B is (2b, 2c) and C is (2d, 2e). M is the midpoint of AB and N is the midpoint of BC. What is the gradient of MN?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A(0, 0), B(8, 2) and C(10, 10) are three corners of a quadrilateral. M is the midpoint of AB and N is the midpoint of BC. What is the gradient of MN?

3. In a quadrilateral ABCD the diagonal AC is 26 units long. M is the midpoint of AB and N is the midpoint of BC. How long is MN?

Level 2 · Building

4. The diagonals of a quadrilateral are 12 and 16 units long and they are perpendicular. The midpoint quadrilateral is a rectangle. What is its area?

5. A(0, 0), B(6, 0), C(8, 6) and D(2, 8) are the corners of a quadrilateral. The midpoints of its sides make a parallelogram, and that parallelogram's diagonals cross at one point. What is the y-coordinate of that point? Give your answer to one decimal place.

Level 3 · Stretch

6. A(1, 2), B(0, 0), C(9, 8) and D(6, k) are the corners of a quadrilateral, with k greater than 0. The midpoints of its sides make a rhombus. What is k?

7. A(0, 0), B(6, 0), C(10, 8) and D(2, 12). P, Q, R and S are the midpoints of AB, BC, CD and DA. The sides QR and PS share the same gradient. What is it?

Something to think about

The midpoint quadrilateral is a rectangle when the diagonals of ABCD are perpendicular. Does that mean ABCD itself is a rectangle? Try to describe an ABCD, not a rectangle, with perpendicular diagonals.

One way to think about it

No. A kite has perpendicular diagonals and is not a rectangle, and neither is a rhombus. The midpoint quadrilateral only 'sees' the diagonals of ABCD (their lengths and directions), so any quadrilateral with perpendicular diagonals gives a rectangle.

Exit ticket

8. The diagonals of a quadrilateral PQRS are both 10 units long, and they are not perpendicular. What is the most specific name that is certain for the quadrilateral made by joining the midpoints of its sides?