069i - Finding the Hypotenuse

Lesson 69 — Finding the Hypotenuse; Guided Practice

Strand: Measurement | Descriptor: AC9M8M06 | Duration: 45 minutes

Learning Intentions

• To apply Pythagoras’ theorem to calculate the length of the hypotenuse given the two legs.
• To express answers as exact surds or as decimals rounded to a specified accuracy, with correct units.

Success Criteria

I can:

1. Substitute correctly into to find the hypotenuse .
2. Simplify a square root to exact surd form when the answer is not a whole number.
3. Round a decimal answer to a stated accuracy, with correct units.
4. Solve applied problems (e.g. a ladder, a screen diagonal) by finding a hypotenuse.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. State Pythagoras’ theorem.
2. In each of three rotated triangle diagrams, identify the hypotenuse.
3. Simplify to exact surd form. (Recall from the Year 8 irrational numbers unit: .)
4. Round to two decimal places.

Answers: 1. , legs , hypotenuse ; 2. Always the side opposite the right angle; 3. ; 4. .

Activities

Activity 1 — Guided Practice: Calculating the Hypotenuse (14 min)

I do:

I do (non-perfect-square case):

We do: Legs and . Give the answer both as an exact surd and rounded to decimal place.

You do:

1. Legs .
2. Legs (exact surd, then rounded to d.p.).
3. Legs (exact surd, then rounded to d.p.).
4. Legs (rounded to d.p.).

Activity 2 — Applied Problems (13 min)

Pairs. Every final answer must carry correct units.

I do: A ladder reaches m up a wall, with its foot placed m from the wall’s base. Find the ladder’s length.

Problem 1. A rectangular gate is m wide and m tall. A diagonal brace is fitted. How long is the brace, to decimal place?

Problem 2. A TV screen has width cm and height cm. Find its diagonal size, to the nearest cm.

Problem 3. A square has side length cm. Find the exact length of its diagonal as a surd, then as a decimal rounded to decimal places.

Activity 3 — Inquiry: Estimate before You Calculate (8 min)

Pairs. Before calculating, estimate whether the hypotenuse of a triangle with legs and will be closer to , , or — without squaring anything.

Teacher prompt: “The hypotenuse must always be longer than either leg, but shorter than the sum of both legs. What range does that give you here?”

Check the estimate against the exact calculation. Repeat for legs and (estimate first: will be closer to , , or ?).

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the hypotenuse of a right triangle with legs and .
2. Find the hypotenuse of a right triangle with legs and , as an exact surd.
3. Round your answer to Q2 to decimal places.
4. A rectangular field is m by m. Find the length of its diagonal path.
5. Reasoning. Explain why the hypotenuse must always be greater than either leg on its own, without calculating anything.

Answers: 1. ; 2. ; 3. ; 4. m; 5. The hypotenuse squared equals the sum of both legs squared, which is always more than either leg squared alone (since all terms are positive) — so the hypotenuse itself must be longer than either individual leg.

Common Misconceptions

Misconception	How to pre-empt it
Stopping at and forgetting to take the square root.	Require every solution to explicitly show the square-root step as its own line.
Adding the legs instead of squaring and adding: .	Contrast directly with a numeric check, e.g. .
Rounding too early (e.g. rounding before taking the root).	Insist on keeping full precision until the final step.
Leaving an answer as an unsimplified surd, e.g. instead of .	Always ask “can this surd be simplified?” as a final check.
Giving a surd answer in a real-world context where a practical decimal is expected (e.g. cutting a physical length of timber).	Discuss context: exact surds suit theoretical/geometric answers; decimals suit measurement/construction answers.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A rectangle has diagonal cm and one side cm. Find the other side.

Answer

cm. (This uses the reverse process — the focus of Lesson 70.)

E2 (Kangaroo style). A cube has edge length cm. Find the length of the diagonal across one face, as an exact surd.

Answer

cm.

E3 (Challenge). Two right-angled triangles share a common leg of length . One has hypotenuse ; the other has hypotenuse . Find the sum of their other two legs.

Answer

First triangle: . Second triangle: . Sum .

Homework

1. Find the hypotenuse: (a) legs (b) legs (c) legs (exact surd) (d) legs (rounded to d.p.).
2. A ladder’s foot is m from a wall, reaching m up. Find the ladder’s length to decimal place.
3. Simplify: (a) (b) (c) .
4. A rectangular sports field is m by m. Find the diagonal distance across it, to the nearest metre.
5. Reasoning. A student calculates a hypotenuse and gets an answer shorter than one of the legs. Explain what error must have occurred.
6. Challenge. An equilateral triangle has side length cm. By splitting it into two right-angled triangles with an altitude, find the exact height as a surd, then round to decimal places.

Answers: 1. (a) (b) (c) (d) ; 2. m; 3. (a) (b) (c) ; 4. m; 5. The hypotenuse must always be the longest side; an answer shorter than a leg means either the wrong side was solved for (a leg was found using the hypotenuse formula) or an arithmetic error occurred, likely subtracting instead of adding the squares; 6. The altitude splits the base into two cm halves: cm.