Computing the missing side

The most useful skill from this chapter is also the simplest: given two sides of a right triangle, you can always find the third in one line.

Concept

Three cases.

1. Find the hypotenuse given the two legs $a, b$: $c = \sqrt{a^2 + b^2}$.

2. Find a leg given the hypotenuse $c$ and the other leg $b$: $a = \sqrt{c^2 - b^2}$.

3. (Trivial , for completeness.) Given hypotenuse $c$ and a leg $b$, the other leg is also $\sqrt{c^2 - b^2}$.

Procedure.

1. Sketch the triangle and label the right angle clearly. The hypotenuse is opposite the right angle and is always the longest side.
2. Decide which of the three cases you are in.
3. Write down the formula, square the known sides, do the arithmetic.
4. Take a (positive) square root.

Common pitfalls.

• Using $c$ as if it were a leg. The hypotenuse is always opposite the right angle.
• Forgetting to square when applying the formula (writing $c = a + b$ , wrong).
• Negative signs when subtracting under the root (means you misidentified which side is the hypotenuse).

Useful shortcut. If the two given sides match a Pythagorean triple (like $3, 4$ or $5, 12$), the answer is the third member of the triple ($5$ or $13$) , no calculation needed.

Decimal answers. Many right triangles don't have integer sides. Don't be alarmed. $\sqrt{2}, \sqrt{5}, \sqrt{50}$ are perfectly valid lengths, even if they have non-terminating decimals.

Worked examples

Example 1. Find $c$ given $a = 6, b = 8$.

• $c = \sqrt{36 + 64} = \sqrt{100} = 10$.

Example 2. Find the leg $a$ given $c = 13, b = 5$.

• $a = \sqrt{169 - 25} = \sqrt{144} = 12$.

Example 3. A right triangle has legs $1$ and $2$. Find the hypotenuse.

• $c = \sqrt{1 + 4} = \sqrt{5} \approx 2.236$.

Example 4. A ladder $5$ m long leans against a wall. Its foot is $3$ m from the wall. How high does the top touch?

• Sketch: right triangle. Ladder is hypotenuse $= 5$. Distance from wall = leg $= 3$.
• Height $= \sqrt{25 - 9} = \sqrt{16} = 4$ m.

Try it yourself

1. Find the hypotenuse if legs are $9$ and $12$.
2. Find the leg if hypotenuse $= 17$ and the other leg is $8$.
3. A rectangle has sides $9$ cm and $12$ cm. Find its diagonal.
4. Find the diagonal of a square of side $10$ cm.
5. A ladder leans against a wall reaching $4$ m up; the foot is $3$ m from the wall. Length of ladder?
6. The hypotenuse of a right triangle is $10$. One leg is $6$. Find the other.
7. Find the hypotenuse if legs are $\sqrt{2}$ and $\sqrt{7}$.
8. Two trees are $6$ m and $11$ m tall, standing $12$ m apart. The distance between their tops is?

Activity

Ladder safety. Building codes for ladders recommend a base-to-wall ratio of about $1 : 4$ (so a ladder leaning a quarter of its length away from the wall is safe). For a $5$ m ladder, the base should be $1.25$ m from the wall. Use the theorem to find the height it reaches: $\sqrt{25 - 1.5625} \approx 4.84$ m. Compare with the height you computed for a "$3$ m base" ladder. Safety is built on geometry.