Converse and a touch of trigonometry

The theorem has a converse: if the sides of a triangle satisfy $a^2 + b^2 = c^2$, then it must be a right triangle. This gives us a powerful way to test whether a triangle is right-angled without measuring any angle.

Concept

Converse of the theorem. Given a triangle with sides $a, b, c$ (and $c$ the longest), if $a^2 + b^2 = c^2$, the triangle is right-angled (and the right angle is opposite $c$).

Examples.

• Sides $5, 12, 13$: $25 + 144 = 169 = 13^2$. So right-angled.
• Sides $6, 7, 9$: $36 + 49 = 85 \ne 81 = 9^2$. Not right-angled.

Distinguishing triangle types from sides.

• $a^2 + b^2 = c^2$: right-angled.
• $a^2 + b^2 > c^2$: acute (all angles $< 90^\circ$).
• $a^2 + b^2 < c^2$: obtuse (one angle $> 90^\circ$ , the one opposite $c$).

For example, sides $5, 6, 8$: $25 + 36 = 61 < 64 = 8^2$. So obtuse , the angle opposite the $8$-side is greater than $90^\circ$.

A glimpse of trigonometry. In a right triangle, beyond the sides themselves, the ratios between sides depend only on the (non-right) angles. We name three ratios:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}.$

These are the sine, cosine, and tangent of the angle $\theta$. You will study them in detail next year. For now just know:

• They are ratios, so they have no units.
• They depend only on the angle, not the size of the triangle.
• The Pythagoras theorem can be written as $\sin^2 \theta + \cos^2 \theta = 1$ , the most-used identity in trigonometry.

Special values.

Why a peek? Trigonometry is the natural sequel to Pythagoras: where Pythagoras measures lengths, trigonometry measures angles by their effect on lengths. Both run on the same right-triangle machinery.

Worked examples

Example 1. Is a triangle with sides $9, 40, 41$ right-angled?

• $81 + 1600 = 1681 = 41^2$. Yes , right-angled.

Example 2. Is a triangle with sides $4, 5, 6$ right, acute, or obtuse?

• $16 + 25 = 41 > 36$. Acute.

Example 3. Is a triangle with sides $5, 6, 9$ valid? If yes, classify it.

• Triangle inequality: $5+6=11 > 9$ ✓. Valid.
• $25 + 36 = 61 < 81$. Obtuse.

Example 4. In a right triangle with $\theta = 45^\circ$, if the hypotenuse is $10$, find the legs.

• $\sin 45^\circ = 1/\sqrt{2}$, so opposite $= 10/\sqrt{2} = 5\sqrt{2}$.
• By symmetry both legs are $5\sqrt{2} \approx 7.07$.

Try it yourself

1. Is the triangle $(7, 24, 25)$ right-angled?
2. Classify the triangle $(6, 8, 11)$ as right/acute/obtuse.
3. Classify the triangle $(8, 15, 17)$.
4. Find the right angle's location in the triangle $(20, 21, 29)$.
5. In a right triangle, one leg is $3$ and the angle opposite it is $30^\circ$. Find the hypotenuse using $\sin 30^\circ$.
6. Show that $\sin^2 30^\circ + \cos^2 30^\circ = 1$.
7. If $\sin \theta = 3/5$, find $\cos \theta$.
8. A right triangle has a $60^\circ$ angle with the side opposite it $= 5\sqrt{3}$. Find the hypotenuse.

Activity / Insight

From Pythagoras to trig. Cut out a right triangle of sides $3, 4, 5$. Measure the smallest angle (opposite the side $3$) with a protractor , it should be near $36.87^\circ$. Compute $\sin(36.87^\circ) = 3/5 = 0.6$, $\cos = 4/5 = 0.8$, $\tan = 3/4 = 0.75$. Check on a calculator. This same correspondence , angle to side-ratio , is what makes trigonometry the natural next step after this chapter.