Analytic Geometry · Analytic Geometry

Verifying Properties of Lines and Triangles

3.2About 70 minutes

Learning goals

I can verify that lines are parallel or perpendicular using slopes.
I can classify a triangle as equilateral, isosceles or scalene using lengths.
I can verify that a triangle is a right triangle using slopes or lengths.
I can write a clear concluding statement for a verification.

Words to know

verify
Show that something is true by calculating and reasoning, not by looking.

equilateral triangle
A triangle with three equal sides.

isosceles triangle
A triangle with at least two equal sides.

scalene triangle
A triangle with no equal sides.

right triangle
A triangle with one 90° angle.

⊥ and ∥
Symbols for "is perpendicular to" and "is parallel to".

From "looks like" to "is"

A triangle drawn on a grid may look isosceles or look like it has a right angle. But looks can fool you: an angle of 88° looks just like 90°. Analytic geometry lets you verify properties exactly, using lengths and slopes calculated from the coordinates.

In this lesson you will verify the type of a triangle, and whether two lines are perpendicular or parallel.

To verify that...	Calculate...	Look for...
two segments are equal	both lengths	equal lengths
two lines are parallel	both slopes	equal slopes
two lines are perpendicular	both slopes	negative reciprocals (product −1), or one horizontal and one vertical
a triangle is equilateral	all three side lengths	all three equal
a triangle is isosceles	all three side lengths	at least two equal
a triangle is right-angled	the slopes of the sides (or the lengths)	two perpendicular sides (or a² + b² = c²)
a point is the midpoint of a segment	the midpoint	the same coordinates

Your analytic geometry toolkit.

Worked example

Example 1: Are the lines perpendicular?

Line 1 passes through (1, −2) and (4, 7). Line 2 passes through (−2, 5) and (4, 3). Verify that the lines are perpendicular.

Slope of line 1: (7 − (−2))/(4 − 1) = 9/3 = 3.
Slope of line 2: (3 − 5)/(4 − (−2)) = −2/6 = −1/3.
3 and −1/3 are negative reciprocals: 3 × (−1/3) = −1.

Therefore the lines are perpendicular.

Worked example

Example 2: An equilateral triangle

Verify that the triangle with vertices A(0, 0), B(6, 0) and C(3, 3√3) is equilateral.

AB = 6 − 0 = 6 (horizontal).
AC = √[(3 − 0)² + (3√3 − 0)²] = √(9 + 9 × 3) = √(9 + 27) = √36 = 6. (Remember (3√3)² = 9 × 3 = 27.)
BC = √[(3 − 6)² + (3√3 − 0)²] = √(9 + 27) = √36 = 6.

AB = AC = BC = 6. Therefore triangle ABC is equilateral.

Worked example

Example 3: Isosceles, and is it right-angled?

Triangle PQR has vertices P(−2, 1), Q(4, 3) and R(2, −3). Classify it by its sides and check whether it has a right angle.

PQ = √(6² + 2²) = √40. QR = √((−2)² + (−6)²) = √40. PR = √(4² + (−4)²) = √32.
PQ = QR, so the triangle is isosceles (and not equilateral, since PR is different).
Slopes: PQ = 2/6 = 1/3; QR = (−6)/(−2) = 3; PR = (−4)/4 = −1.
Check each pair: (1/3)(3) = 1, (1/3)(−1) = −1/3, (3)(−1) = −3. No product is −1, so no two sides are perpendicular.
Check with lengths: the longest sides are √40, and 40 + 32 = 72 ≠ 40, so the Pythagorean relationship fails too ✓.

Triangle PQR is isosceles (PQ = QR = √40) but not a right triangle.

Triangle PQR. The tick marks show the equal sides PQ = QR = √40. The third side PR = √32 is shorter.

Worked example

Example 4: A right triangle

Verify that triangle DEF with D(−3, 2), E(1, 4) and F(3, 0) is a right isosceles triangle.

Slope DE = (4 − 2)/(1 − (−3)) = 2/4 = 1/2. Slope EF = (0 − 4)/(3 − 1) = −4/2 = −2.
(1/2)(−2) = −1, so DE ⊥ EF: there is a right angle at E.
Lengths: DE = √(16 + 4) = √20. EF = √(4 + 16) = √20. So DE = EF.
Check with the Pythagorean relationship: DF = √(36 + 4) = √40, and DE² + EF² = 20 + 20 = 40 = DF² ✓.

Therefore triangle DEF is a right isosceles triangle, with the right angle at E.

The slope triangles show DE has slope 2/4 = 1/2 and EF has slope −4/2 = −2. The slopes are negative reciprocals, so the sides meet at a right angle at E. Each grid square is 1 unit.

Write a concluding statement

An analytic proof is not finished until you say what you have shown and why. Use a sentence like: "Since the slopes of DE and EF are negative reciprocals, DE ⊥ EF, so angle E is a right angle. Therefore triangle DEF is a right triangle." The symbol ⊥ means "is perpendicular to" and ∥ means "is parallel to".

Worked example

Example 5: Verifying a point is on a right bisector

Verify that the point (5, 0) is equidistant from A(1, 3) and B(8, 4). What does that tell you?

Distance to A: √[(5 − 1)² + (0 − 3)²] = √(16 + 9) = √25 = 5.
Distance to B: √[(5 − 8)² + (0 − 4)²] = √(9 + 16) = √25 = 5.

Both distances are 5, so (5, 0) is equidistant from A and B. Therefore it lies on the right bisector of AB.

Just for fun

Why are equilateral triangles so easy to get along with?

They treat every side equally.

Hands-on activity

Grid-point triangles

You need: graph paper, a ruler, a pencil, and the corner of a book or index card (a right-angle tester).

Choose three grid points that you think make an isosceles triangle that is not right-angled. Draw it.
Verify with lengths. If it is not isosceles, adjust one vertex and try again.
Choose three grid points for a right triangle with no horizontal or vertical side. Draw it and check the right angle with the book corner.
Verify with slopes that two sides are perpendicular.
Challenge: can you find three grid points that form an equilateral triangle? Try a few, measuring the lengths.

What to look for: sample answers: (0, 0), (4, 1), (1, 4) is isosceles (two sides √17) and not right; (0, 0), (2, 1), (1, 3) has slopes 1/2 and −2, so a right angle at (2, 1). For step 5: it is impossible to find an equilateral triangle with all three vertices on grid points, so every attempt will be slightly off. (Proving this is beyond this course, but it is a famous result.)

Your turn

Verify that the line through (0, 3) and (4, 5) is parallel to the line through (−2, −1) and (2, 1).

Verify that the line through (2, 1) and (5, 7) is perpendicular to the line through (0, 4) and (6, 1).

Classify triangle A(1, 1), B(5, 1), C(3, 5) by its sides.

Verify that triangle J(0, 0), K(4, 2), L(−1, 2) is a right triangle.

Classify triangle (−3, 0), (3, 0), (0, 5) by its sides and angles.

Verify that (2, −1) is equidistant from (−2, 2) and (5, 3).

More practice: warm-up

What does ⊥ mean?

Two slopes are 4/5 and −5/4. Are the lines perpendicular?

Two slopes are 2 and −2. Are the lines perpendicular?

How many sides of an isosceles triangle are equal?

A triangle has sides √8, √8 and 4. Is it a right triangle?

More practice: on your own

Classify triangle (1, 2), (4, 6), (8, 3) by sides and angles.

Verify that triangle (−4, 0), (0, 3), (3, −1) is a right triangle. Where is the right angle?

Is the triangle (0, 0), (4, 0), (2, 3) equilateral? Explain.

Show that A(−1, −2), B(1, 4) and C(4, 13) are collinear using slopes.

Find k so that the line through (0, 0) and (k, 6) is perpendicular to the line through (1, 1) and (4, 3).

Challenge

Triangle ABC has A(−2, 3), B(4, 5) and C(6, −1). Verify that it is a right triangle, find its area, and find the length of the altitude to the hypotenuse.

Find all values of y so that triangle with vertices (0, 0), (6, 0) and (3, y) is equilateral. Explain why the answer is not a whole number.

Point P(a, b) is equidistant from A(0, 0) and B(4, 0) and is 5 units from A. Find all possible P.

Check your understanding

Quiz

Two lines have slopes −3/4 and 4/3. The lines are:

To verify that a triangle is isosceles, you should calculate:

Which triangle is a right triangle?

Triangle sides are √18, √18 and 6. What is it?

What does it mean if the slopes of AB and BC are equal?

Which completes the proof? "Slope DE = 1/2 and slope EF = −2, so ___ "

For parents and teachers

Big idea: lengths verify equal sides; slopes verify parallel or perpendicular lines. A verification ends with a concluding sentence. This lesson teaches expectation 3.2 (verify, using algebraic techniques and analytic geometry, characteristics of geometric figures, such as perpendicular lines and equilateral triangles), and practises the reasoning-and-proving and communicating processes.

How to help: Ask your teen to finish every problem with a "Therefore..." sentence that names the property and the evidence. Draw the figure first; it helps decide what to calculate.

Common difficulties:

Relying on how a diagram looks.
Using slopes to verify equal lengths, or lengths to verify parallel lines.
Squaring expressions like 3√3 (it is 9 × 3 = 27, not 27√3 or 9√3).
Forgetting that a horizontal and a vertical line are perpendicular even though "product −1" does not apply (a vertical line has no slope).