Analytic Geometry · Analytic Geometry

Investigating Properties of Triangles and Quadrilaterals

3.1About 70 minutes

Learning goals

I can investigate geometric properties using paper folding, measurement, coordinates and dynamic geometry software.
I can make a conjecture and test it, including looking for counterexamples.
I can describe properties of medians and the centroid of a triangle.
I can describe how areas of similar figures on the sides of a right triangle are related.

Words to know

conjecture
A statement believed to be true, based on examples, that has not yet been proved.

counterexample
An example that shows a statement is false.

centroid
The point where the three medians of a triangle meet; the balance point.

similar figures
Figures with the same shape: corresponding angles equal and corresponding lengths in proportion.

diagonal
A segment joining two vertices of a polygon that are not next to each other.

dynamic geometry software
Software (such as GeoGebra) for constructing figures that can be measured and dragged.

Mathematicians as detectives

Before anyone can prove a geometric property, someone has to notice it. Mathematicians investigate: they draw, fold, measure and experiment, look for patterns, and make a conjecture (an educated guess about what is always true). Then they test it on more examples, look for a counterexample, and finally try to verify or prove it.

In this lesson you will investigate properties of triangles and quadrilaterals with paper folding, measuring, coordinates and dynamic geometry software. In the next lessons you will verify them with analytic geometry.

Remember from Grade 9

In Grade 9 you learned that the three medians of a triangle meet at the centroid, the triangle's balance point, which is two-thirds of the way along each median from its vertex. You also met the circumcentre (where the perpendicular bisectors meet). Now you will check these facts with coordinates.

Tools for investigating

Paper folding: fold a vertex onto another to find a midpoint or a right bisector; fold a side onto itself through a vertex to make an altitude.
Measuring: a ruler and protractor on graph paper.
Dynamic geometry software: free programs such as GeoGebra let you construct a figure, measure lengths, slopes and angles, and then drag the vertices. If a property stays true while you drag, it is probably true for every figure of that type. If it breaks, you have found a counterexample.
Coordinates: calculate midpoints, lengths and slopes for a specific example.

Worked example

Example 1: Medians and the centroid

Triangle ABC has vertices A(−4, −2), B(6, 0) and C(1, 8). Investigate its medians. Do they meet at one point? Where along each median is that point?

Midpoints: AB → (1, −1); BC → (3.5, 4); AC → (−1.5, 3).
The median from C goes from (1, 8) to (1, −1): it is the vertical line x = 1.
The median from A goes from (−4, −2) to (3.5, 4): slope 6/7.5 = 0.8, so y = 0.8x + 1.2. At x = 1: y = 2. So it crosses the first median at G(1, 2).
The median from B goes from (6, 0) to (−1.5, 3): slope 3/(−7.5) = −0.4, so y = −0.4x + 2.4. At x = 1: y = 2 ✓. All three medians pass through G(1, 2).
Along the median from C: CG = 8 − 2 = 6 and G to (1, −1) is 2 − (−1) = 3. So CG : GM = 6 : 3 = 2 : 1.
Notice also: ((−4 + 6 + 1)/3, (−2 + 0 + 8)/3) = (3/3, 6/3) = (1, 2). The centroid is the average of the three vertices.

The medians meet at the centroid G(1, 2), which divides each median in the ratio 2 : 1 from the vertex. This is one example; the next lessons show how to verify such properties in general.

The three dashed medians of triangle ABC all pass through the centroid G(1, 2). The ringed dots are the midpoints of the sides. G is twice as far from each vertex as from the midpoint of the opposite side.

Similar figures on the sides of a right triangle

You know that the areas of squares drawn on the sides of a right triangle satisfy a² + b² = c². Is that only true for squares? Investigate with other shapes. Shapes are similar if one is an enlargement or reduction of the other.

Worked example

Example 2: Semicircles on a 6-8-10 triangle

A right triangle has legs 6 and 8 and hypotenuse 10. A semicircle is drawn on each side, with that side as its diameter. Compare the areas.

Area of a semicircle with diameter d: ½π(d/2)² = πd²/8.
On the leg 6: π(36)/8 = 4.5π. On the leg 8: π(64)/8 = 8π. On the hypotenuse 10: π(100)/8 = 12.5π.
4.5π + 8π = 12.5π ✓.
Why? Every area here is (π/8) × (side)². Since 6² + 8² = 10², multiplying both sides by π/8 keeps them equal.

The semicircles on the two legs have the same total area as the semicircle on the hypotenuse. The same is true for any similar figures (equilateral triangles, regular hexagons...) drawn on the three sides, because each area is a constant times the side squared.

Semicircles on the sides of a right triangle with legs 6 and 8 and hypotenuse 10 (drawn to scale). The two smaller areas add up to the largest: 4.5π + 8π = 12.5π.

Quadrilaterals: what do the diagonals do?

A diagonal joins two opposite vertices. Investigating the diagonals is a powerful way to tell quadrilaterals apart. The table summarizes properties you can discover by drawing and measuring (or by dragging in GeoGebra).

Quadrilateral	Sides	Diagonals
Parallelogram	Opposite sides parallel and equal	Bisect each other
Rectangle	Parallelogram with four right angles	Bisect each other and are equal in length
Rhombus	Parallelogram with four equal sides	Bisect each other and are perpendicular
Square	Rectangle and rhombus	Bisect each other, equal, perpendicular
Kite	Two pairs of equal adjacent sides	Perpendicular; one bisects the other
Trapezoid	Exactly one pair of parallel sides (in this course)	No special property in general

Properties of quadrilaterals that you can investigate and later verify.

Worked example

Example 3: Join the midpoints of any quadrilateral

Quadrilateral ABCD has vertices A(0, 0), B(8, −2), C(11, 6) and D(1, 7). Join the midpoints of its sides in order. What kind of shape do you get?

Midpoints: AB → (4, −1); BC → (9.5, 2); CD → (6, 6.5); DA → (0.5, 3.5).
Slope from (4, −1) to (9.5, 2): 3/5.5 = 6/11. Slope from (0.5, 3.5) to (6, 6.5): 3/5.5 = 6/11. Parallel.
Slope from (9.5, 2) to (6, 6.5): 4.5/(−3.5) = −9/7. Slope from (4, −1) to (0.5, 3.5): 4.5/(−3.5) = −9/7. Parallel.
Both pairs of opposite sides are parallel, so the midpoint shape is a parallelogram.
Conjecture: joining the midpoints of the sides of any quadrilateral makes a parallelogram. Test it: draw a few very different quadrilaterals (including a very skinny one) and check. In GeoGebra, drag the vertices and watch the inner shape.

The midpoint quadrilateral is a parallelogram. (This is called Varignon's theorem, after the French mathematician Pierre Varignon. It follows from the midsegment property you will verify in the last lesson.)

ABCD is an irregular quadrilateral, but the shape formed by joining the midpoints of its sides (the inner shaded shape) has both pairs of opposite sides parallel: it is a parallelogram.

Conjecture, test, counterexample

One example never proves a conjecture is always true, but one counterexample proves it is false. Example: "The diagonals of a parallelogram are equal." Test the parallelogram with vertices (0, 0), (4, 0), (6, 3), (2, 3): the diagonals are √45 and √13, which are not equal. So the conjecture is false. (It is true for rectangles.)

Just for fun

A conjecture is like a rumour in math class: very interesting, but nobody believes it until it has been proved.

Hands-on activity

Balance a triangle on its centroid

You need: a piece of thin cardboard (a cereal box works), a ruler, a pencil, scissors and a pencil with an eraser end. Grown-up note: a grown-up should help with cutting cardboard.

Draw a large triangle (each side at least 15 cm) on the cardboard and cut it out.
Find the midpoint of each side with the ruler and mark it.
Draw the three medians (each vertex to the midpoint of the opposite side). Do they meet at one point? Label it G.
Measure one median: is the distance from the vertex to G about twice the distance from G to the midpoint?
Try to balance the triangle flat on the eraser end of a pencil at G. Then try a point that is not G.
Bonus: put the triangle on graph paper, record the vertex coordinates, and check that G is close to their average.

What to look for: the medians meet at one point (small drawing errors are normal); the vertex-to-G part is about twice the G-to-midpoint part; the triangle balances at G and tips over elsewhere. The centroid is the triangle's centre of mass when the cardboard is uniform.

Your turn

Find the centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9).

For that triangle, check the 2 : 1 ratio on the median from (0, 9).

Equilateral triangles are drawn on the sides of a right triangle with legs 5 and 12. The triangle on the leg 5 has area about 10.83 and on the leg 12 about 62.35. Predict the area of the one on the hypotenuse.

Which quadrilaterals have diagonals that are equal in length?

Which quadrilaterals have perpendicular diagonals?

Find a counterexample to: "If a quadrilateral has four equal sides, it is a square."

More practice: warm-up

What is a conjecture?

What is a counterexample?

Where do the three medians of a triangle meet?

What do the diagonals of every parallelogram do?

Name one free piece of dynamic geometry software.

More practice: on your own

Find the centroid of the triangle (−2, 5), (4, −1), (7, 8).

Squares are drawn on the sides of a right triangle. The squares on the legs have areas 49 cm² and 576 cm². What is the area of the square on the hypotenuse and the length of the hypotenuse?

Join the midpoints of the sides of the rectangle (0, 0), (8, 0), (8, 4), (0, 4). What shape do you get? Explain using lengths.

Test the conjecture "The diagonals of a rhombus are equal" on the rhombus (0, 0), (5, 0), (8, 4), (3, 4).

Describe how to use paper folding to find the centroid of a paper triangle.

Challenge

Investigate: in triangle A(0, 0), B(8, 0), C(2, 6), find the centroid G, the circumcentre O(4, 2), and the orthocentre H (where the altitudes meet, which is (2, 2) here). Show that O, G and H lie on one line, and that GH = 2·OG.

Regular hexagons are drawn on the sides of a right triangle with sides 3, 4 and 5. Explain without calculating why the two smaller hexagons together have the same area as the largest.

Make and test a conjecture about the shape formed by joining the midpoints of the sides of a rhombus.

Check your understanding

Quiz

What is a counterexample?

Where is the centroid of the triangle (0, 0), (9, 0), (3, 6)?

The centroid divides each median in the ratio (from the vertex):

Semicircles on the legs of a right triangle have areas 9π and 16π. What is the area of the semicircle on the hypotenuse?

Which quadrilateral's diagonals are always perpendicular?

Joining the midpoints of the sides of any quadrilateral gives a:

For parents and teachers

Big idea: investigate first (fold, measure, drag in software, calculate examples), make a conjecture, test it and look for counterexamples. This lesson teaches expectation 3.1 (determine, through investigation, characteristics and properties of geometric figures, such as medians in a triangle and similar figures on the sides of a right triangle). It also practises the reasoning-and-proving process.

How to help: The cardboard balancing activity is memorable: the triangle really does balance at the centroid. If you have a computer, explore GeoGebra together; dragging a vertex and watching a measurement stay the same is convincing.

Common difficulties:

Believing one example proves a statement. Ask, "Could there be a shape where this fails?"
Mixing up the properties of rectangles (equal diagonals) and rhombuses (perpendicular diagonals).
Drawing "general" examples that are accidentally special (e.g. a quadrilateral that is secretly a parallelogram). Encourage irregular shapes when testing.