Analytic Geometry · Analytic Geometry

Verifying Properties of Quadrilaterals

3.2 3.3About 75 minutes

Learning goals

I can classify a quadrilateral as a parallelogram, rectangle, rhombus, square, trapezoid or kite using slopes and lengths.
I can verify properties of diagonals using midpoints, lengths and slopes.
I can choose which properties to calculate to classify a quadrilateral efficiently.

Words to know

parallelogram
A quadrilateral with both pairs of opposite sides parallel.

rectangle
A parallelogram with four right angles.

rhombus
A parallelogram with four equal sides.

square
A quadrilateral that is both a rectangle and a rhombus.

trapezoid
A quadrilateral with exactly one pair of parallel sides (in this course).

kite
A quadrilateral with two pairs of equal adjacent sides.

bisect each other
Two segments cut each other into equal halves; they share a midpoint.

Is it really a square?

A new skateboard park is laid out on a city grid. The plan calls for a square concrete pad, and the contractor has marked four corners with stakes. How can the inspector be sure the pad is truly a square, and not a rhombus or a rectangle that is slightly off? With the coordinates of the four corners, analytic geometry can answer exactly.

In this lesson you will verify the type of a quadrilateral and properties of its diagonals.

The family of quadrilaterals. Each shape has all the properties of the shapes above it that it is joined to. A square is a rectangle and a rhombus, and both are parallelograms.

What to calculate

Name the vertices in order around the shape: ABCD means sides AB, BC, CD and DA, and diagonals AC and BD.

Parallelogram: both pairs of opposite sides are parallel (slope AB = slope DC and slope BC = slope AD). Or: the diagonals have the same midpoint (they bisect each other).
Rectangle: a parallelogram with a right angle (adjacent slopes are negative reciprocals). Or: a parallelogram with equal diagonals.
Rhombus: all four sides are equal. Or: a parallelogram with perpendicular diagonals.
Square: a rhombus with a right angle (four equal sides and adjacent sides perpendicular).
Trapezoid: exactly one pair of opposite sides is parallel. (Some books say "at least one pair"; in this course we use "exactly one".)
Kite: two pairs of equal adjacent sides (AB = AD and CB = CD, for example).

Plan before you calculate: decide what you need to show, then compute only what you need.

Worked example

Example 1: A parallelogram, but not a rectangle or rhombus

Quadrilateral ABCD has vertices A(−3, −1), B(3, 1), C(5, 5) and D(−1, 3). Classify it as precisely as you can.

Slopes of the sides: AB = 2/6 = 1/3; BC = 4/2 = 2; DC = 2/6 = 1/3; AD = 4/2 = 2.
AB ∥ DC (both 1/3) and BC ∥ AD (both 2), so ABCD is a parallelogram.
Right angle? (1/3)(2) = 2/3 ≠ −1, so AB is not perpendicular to BC: not a rectangle.
Equal sides? AB = √(36 + 4) = √40 and BC = √(4 + 16) = √20. Not equal: not a rhombus.

ABCD is a parallelogram that is not a rectangle or a rhombus.

Worked example

Example 2: A rhombus

Verify that P(0, 0), Q(4, 3), R(9, 3) and S(5, 0) form a rhombus, and that its diagonals are perpendicular. Is it a square?

Side lengths: PQ = √(16 + 9) = 5; QR = 5 (horizontal); RS = √(16 + 9) = 5; SP = 5 (horizontal). All four sides are 5, so PQRS is a rhombus.
Diagonals: slope PR = 3/9 = 1/3; slope QS = (0 − 3)/(5 − 4) = −3. (1/3)(−3) = −1, so the diagonals are perpendicular ✓.
Square? Slope PQ = 3/4 and QR is horizontal (slope 0). These are not perpendicular, so the angles are not 90°.

PQRS is a rhombus with perpendicular diagonals, but not a square.

Worked example

Example 3: The diagonals of a rectangle bisect each other

Quadrilateral KLMN has vertices K(1, 1), L(7, 4), M(5, 8) and N(−1, 5). Verify that it is a rectangle, and that its diagonals bisect each other and are equal in length.

Slopes: KL = 3/6 = 1/2; LM = 4/(−2) = −2; MN = (−3)/(−6) = 1/2; NK = (−4)/2 = −2.
Opposite sides are parallel (1/2 and 1/2; −2 and −2), so KLMN is a parallelogram. Adjacent slopes multiply to (1/2)(−2) = −1, so the angles are right angles. KLMN is a rectangle.
Midpoints of the diagonals: KM: ((1 + 5)/2, (1 + 8)/2) = (3, 4.5). LN: ((7 + (−1))/2, (4 + 5)/2) = (3, 4.5). The same point, so the diagonals bisect each other.
Lengths of the diagonals: KM = √(16 + 49) = √65. LN = √(64 + 1) = √65. Equal.

KLMN is a rectangle; its diagonals bisect each other at (3, 4.5) and are equal (√65 each).

Rectangle KLMN. Both dashed diagonals have midpoint (3, 4.5), so they cut each other in half, and both have length √65.

Worked example

Example 4: Trapezoid or parallelogram?

Classify the quadrilateral W(−2, 0), X(6, 0), Y(4, 4), Z(0, 4).

Slopes: WX = 0; ZY = 0, so WX ∥ ZY.
XY = 4/(−2) = −2; WZ = 4/2 = 2. Not equal, so XY is not parallel to WZ.
Exactly one pair of parallel sides: a trapezoid.
Bonus: WZ = √(4 + 16) = √20 and XY = √(4 + 16) = √20. The non-parallel sides are equal, so it is an isosceles trapezoid.

WXYZ is an isosceles trapezoid.

Be efficient, be complete

To show a shape is a square, it is enough to show it is a rhombus (four equal sides) and has one right angle. To show it is not a square, one failed test is enough. Always state what each calculation shows, and finish with a concluding sentence.

Just for fun

A square walks into a party of quadrilaterals. The rectangles cheer, "You're one of us!" The rhombuses cheer, "No, you're one of us!" The parallelograms just smile and say, "Relax, everyone. We're all family here."

Hands-on activity

Pencil diagonals

You need: two pencils of the same length, one pencil (or straw) of a different length, a sheet of paper, and a ruler.

Mark the middle of each pencil with a small dot (measure with the ruler).
Cross the two equal pencils on the paper so their middles touch, at any angle. Mark the four pencil tips on the paper and join them with the ruler. What shape do you get?
Repeat with the two equal pencils crossed at a right angle (use the corner of a sheet of paper to check). What shape now?
Cross one long and one short pencil at their middles, at a right angle. Join the tips. What shape?
Cross them at their middles, but not at a right angle. What shape?
Write a rule: what do the diagonals tell you about the quadrilateral?

What to look for: equal diagonals bisecting each other give a rectangle; equal and perpendicular give a square; unequal and perpendicular (bisecting) give a rhombus; unequal and not perpendicular (bisecting) give a parallelogram.

Your turn

Verify that A(1, 2), B(5, 3), C(6, 7), D(2, 6) form a parallelogram.

Is the parallelogram in the last question a rhombus?

Show that E(0, 0), F(3, 1), G(2, 4), H(−1, 3) is a square.

For the square in the last question, verify that the diagonals bisect each other and are perpendicular.

Classify the quadrilateral (0, 0), (8, 0), (6, 3), (1, 3).

A kite has vertices A(0, 3), B(2, 0), C(0, −5), D(−2, 0). Verify the two pairs of equal adjacent sides.

More practice: warm-up

What must be true about the diagonals of a parallelogram?

A parallelogram has a right angle. What is it?

A quadrilateral has four equal sides. What is it?

What two properties together make a square?

In quadrilateral ABCD, name the two diagonals.

More practice: on your own

Show that the diagonals of the parallelogram A(−3, −1), B(3, 1), C(5, 5), D(−1, 3) bisect each other.

Classify P(−2, 1), Q(2, −1), R(4, 3), S(0, 5) as precisely as possible.

Verify that the quadrilateral (1, 1), (6, 1), (9, 5), (4, 5) is a rhombus but not a square.

Three vertices of a rectangle are (−2, 1), (4, 1) and (4, 5). Find the fourth vertex and verify that the diagonals are equal.

Is the quadrilateral (0, 0), (5, 1), (6, 6), (1, 5) a rhombus, a square, or neither? Use the diagonals.

Challenge

Find the value of k so that A(0, 0), B(6, 2), C(k, 7) and D(1, 5) is a parallelogram.

The skateboard pad has corners at (2, 1), (10, 3), (8, 11) and (0, 9) (metres). Is it a square? If not, what is it?

Prove that if the diagonals of quadrilateral ABCD have the same midpoint, then ABCD is a parallelogram. Use A(0, 0), C(2a, 2b), and B(p, q), with the common midpoint M(a, b).

Check your understanding

Quiz

Quadrilateral ABCD has slope AB = slope DC and slope BC = slope AD. It must be a:

The diagonals of a quadrilateral bisect each other and are perpendicular but not equal. It is a:

How can you show diagonals AC and BD bisect each other?

Which set of facts proves that ABCD is a square?

A quadrilateral has exactly one pair of parallel sides. In this course it is called a:

Rectangle diagonals are always:

For parents and teachers

Big idea: quadrilaterals are classified by parallel sides (slopes), equal sides (lengths), right angles (perpendicular slopes) and the behaviour of their diagonals (midpoints, lengths, slopes). This lesson continues expectation 3.2 (verify characteristics of geometric figures) and begins expectation 3.3 (plan and carry out a multi-step strategy, such as verifying that the diagonals of a rectangle bisect each other, which is the curriculum's own example).

How to help: Ask your teen to decide before calculating which properties they need to check. The pencil-diagonals activity makes the diagonal rules visible.

Common difficulties:

Naming vertices out of order (ABCD must go around the shape), which mixes up sides and diagonals.
Stopping too early: showing a shape is a parallelogram does not show it is a rectangle.
Thinking every rhombus is a square, or every rectangle is a square.
Trapezoid definitions differ between books; this unit uses "exactly one pair of parallel sides".