073i - Problem Solving with Right-Angled Triangles

Lesson 73 — Problem Solving with Right-Angled Triangles

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

Learning Intentions

• To use the converse of Pythagoras’ theorem to test whether a triangle is right-angled.
• To solve multi-step problems that combine Pythagoras’ theorem with other measurement and geometry skills.

Success Criteria

I can:

1. Test whether a triangle with three given side lengths is right-angled, using the converse of Pythagoras’ theorem.
2. Correctly identify the longest side as the candidate hypotenuse before testing.
3. Solve multi-step problems combining Pythagoras’ theorem with perimeter, area, or shape properties.
4. Explain my reasoning and check that a solution is realistic.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. If you know two side lengths of a right-angled triangle, you can find the third.
2. The hypotenuse is the longest side of a right-angled triangle.
3. , , is the only whole-number combination that forms a right-angled triangle.
4. Doubling all three sides of a right-angled triangle keeps it right-angled.

Answers: 1. Always — Pythagoras’ theorem always determines the third side. 2. Always — by definition, the side opposite the right angle is the longest. 3. Never — there are infinitely many Pythagorean triples, e.g. and . 4. Always — scaling every side by the same factor preserves all angles.

Activities

Activity 1 — Explicit Instruction: the Converse of Pythagoras’ Theorem (15 min)

Remind students: Pythagoras’ theorem works in both directions. If for the three sides of a triangle (with the longest), the triangle must contain a right angle.

I do: Test whether a triangle with sides , , is right-angled. The longest side, , is the candidate hypotenuse.

Since the two sides match, the triangle is right-angled.

We do: Test , , .

Close, but not equal — this triangle is not right-angled. There is no “close enough” in this test.

You do: Test each set of sides. State whether each triangle is right-angled, showing full working.

(Answers: 1. Yes () 2. No () 3. Yes () 4. Yes () 5. No ())

Activity 2 — Applied Multi-step Problems (18 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. A farmer measures the diagonal of a rectangular field, m by m, to check that a corner is a true right angle. She measures the diagonal as m. Is the corner square? Explain using the converse of Pythagoras’ theorem.

Problem 2. A wheelchair ramp rises m over a horizontal run of m. Find the length of the ramp’s sloped surface, correct to decimal places.

Problem 3. A rhombus has diagonals of length cm and cm. (Recall: the diagonals of a rhombus bisect each other at right angles.) Find the perimeter of the rhombus.

Problem 4 (hardest). Builders use a well-known trick to check that a wall meets the floor at a true right angle, without any angle-measuring tool: mark a point m up the wall and a point m along the floor from the same corner, then measure the distance between the two marks directly. A builder does this and measures m between the marks. Is the wall vertical?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand the problem	We are not finding a missing side — we are testing whether an angle is exactly , using only lengths.
What would the diagonal measurement be if the corner were a true right angle?	Calculate the expected value using Pythagoras’ theorem directly.
Devise a plan	Compute the expected diagonal for legs m and m, then compare it with the builder’s actual measurement of m.
Carry out the plan	m expected, versus m measured.
Looking back	Since , the corner is not exactly a right angle — the wall leans slightly. Is a m difference significant at this scale? (Yes — over only – m, that is a large deviation.)

Answers: 1. , so yes, the corner is square. 2. m. 3. Each side cm, so perimeter cm. 4. Not vertical — see scaffolding above.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Are , , the side lengths of a right-angled triangle? Show your check.
2. A triangle has sides cm, cm, cm. Is it right-angled? Explain.
3. A rhombus has diagonals cm and cm. Find its perimeter.
4. Reasoning. Why must you square and compare the numbers, rather than just checking whether three side lengths “look like” a Pythagorean triple?

Answers: 1. — yes, right-angled. 2. , but ; , so not right-angled. 3. Each side cm, perimeter cm. 4. Because “looking similar” is not proof — only equal squared values guarantee a right angle; two numbers that seem close (like and in the Warmup/Activity 1) can still fail the test.

Common Misconceptions

Misconception	How to pre-empt it
Assuming any three “reasonable-looking” whole numbers form a right-angled triangle.	Always show the full squared comparison — never estimate or eyeball it.
Squaring and adding the two smaller numbers, but comparing against the wrong (non-largest) side.	Always identify and label the longest side as the hypotenuse candidate before starting the check.
Using half-diagonals as if they were full diagonals in a rhombus problem.	Draw the rhombus with its diagonals first, and mark the bisected (halved) lengths explicitly before applying Pythagoras’ theorem.
Accepting a “close enough” result (e.g. vs ) as proof of a right angle.	Emphasise that Pythagoras’ theorem’s converse requires exact equality — no tolerance, unlike real-world measurement error.
Rounding too early in a multi-step problem, causing the final converse check to fail incorrectly.	Keep exact values (or extra decimal places) until the final comparison step.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Which of these are Pythagorean triples: , , ?

Answer

✓; ✗; ✓. So and are Pythagorean triples; is not.

E2 (Kangaroo style). For any whole number , a triangle has sides , , and . Show that this triangle is always right-angled.

Answer

Since the sum of the squares of the two shorter sides always equals the square of , the triangle is right-angled for every whole number — this is a triple generator.

E3 (Challenge). Using the generator from E2, find the Pythagorean triple produced when .

Answer

, , , giving the triple — the same triple as E1!

Homework

1. Test whether each triangle is right-angled: (a) (b) (c) .
2. A rectangular gate’s diagonal is measured as m, and its sides are known to be m and m. Is the gate frame square? Explain.
3. A rhombus has diagonals cm and cm. Find its perimeter.
4. A ramp rises m over a horizontal run of m. Find the sloped length of the ramp, to decimal places.
5. Reasoning. A student says: “If two sides of a triangle are and , the third side must be .” Explain what is wrong with this statement.
6. Challenge. Using the generator triangle , , from the Enrichment section, find the value of that produces the triple , and verify it algebraically.

Answers: Q1 — (a) , yes (b) , no (c) , yes. Q2 — expected diagonal m, which matches, so yes, the gate is square. Q3 — sides cm, perimeter cm. Q4 — m. Q5 — the third side is only if the triangle is right-angled with and as the two legs; without that condition, the third side could be any length satisfying the triangle inequality, and even if it is a right triangle, could be a leg (with as the hypotenuse) rather than the hypotenuse itself. Q6 — ; then and , and indeed .