074i - Consolidation Pythagoras' Theorem

Lesson 74 — Consolidation and Check: Pythagoras’ Theorem

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

Learning Intentions

• To consolidate all applications of Pythagoras’ theorem covered in this unit: finding the hypotenuse, finding a shorter side, composite figures, real-world contexts, and the converse.
• To identify personal areas of strength and weakness ahead of further assessment.

Success Criteria

I can:

1. Find the hypotenuse or a shorter side of a right-angled triangle.
2. Apply Pythagoras’ theorem within composite and three-dimensional figures.
3. Model a real-world scenario as a right-angled triangle and solve it.
4. Use the converse of Pythagoras’ theorem to test for a right angle.
5. Select the correct strategy for an unfamiliar problem without being told which skill to use.

Warmup

(6 minutes — diagnostic quick-fire, mini whiteboards)

1. Find the hypotenuse of a right-angled triangle with legs cm and cm.
2. Find the missing leg of a right-angled triangle with hypotenuse cm and one leg cm.
3. Are , , the sides of a right-angled triangle?
4. A ladder leans against a wall. Which side is the hypotenuse?

Answers: 1. cm; 2. cm; 3. Yes, ; 4. The ladder itself.

Activities

Activity 1 — Skill Review Circuit (15 min)

Individually, then compare with a partner. Four short problems, one from each skill area covered this unit.

Q1 (hypotenuse). Find the hypotenuse of a triangle with legs cm and cm.

Q2 (shorter side). A right-angled triangle has hypotenuse cm and one leg cm. Find the other leg.

Q3 (composite). An isosceles triangle has base cm and equal sides cm. Find its height.

Q4 (converse). Test whether , , form a right-angled triangle.

Answers: Q1 — cm. Q2 — cm. Q3 — half-base , cm. Q4 — , yes.

Activity 2 — Consolidation Problem Set (18 min)

Pairs. Draw a diagram for every problem before calculating.

Problem 1. A rectangular gate measures m by m. Find the length of its diagonal brace.

Problem 2. A rectangular prism-shaped box measures cm cm cm. Find the length of its longest internal diagonal, to decimal place.

Problem 3. A surveyor checks a building’s corner using two marks, m up a wall and m along the ground, and measures m between them. Is the corner square?

Problem 4 (hardest). A kite is flown on a m string. At one moment, the kite is directly downwind of the flyer and m above the ground. How far (horizontally) has the kite drifted from the flyer, assuming the string is pulled taut in a straight line?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand the problem	The string is the hypotenuse of a right-angled triangle; the height and horizontal drift are the two legs.
What do we already know?	Hypotenuse m, one leg (height) m.
Devise a plan	Rearrange Pythagoras’ theorem to find the missing leg.
Carry out the plan	, so m.
Looking back	Check: ✓. Does a –– triple look sensible against known triples (it’s )?

Answers: 1. m. 2. cm. 3. Expected m, but measured m — not equal, so the corner is not exactly square. 4. m.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the hypotenuse of a triangle with legs cm and cm.
2. Find the missing leg of a triangle with hypotenuse cm and one leg cm.
3. A rectangular prism measures cm cm cm. Find its space diagonal.
4. Test whether , , form a right-angled triangle.
5. Reasoning. A student solves a ladder problem and gets a height of m. Explain what has gone wrong.

Answers: 1. cm; 2. cm; 3. cm; 4. , no; 5. A length cannot be negative — the student has likely subtracted the values in the wrong order (e.g. computed instead of ), or mixed up which side is the hypotenuse.

Common Misconceptions

A summary of the biggest misconceptions from across this unit.

Misconception	How to pre-empt it
Using to find a shorter side without rearranging (subtracting instead of adding).	Always identify first whether the hypotenuse or a shorter side is missing, before choosing to add or subtract.
Using the full base instead of half the base in isosceles-triangle or rhombus problems.	Sketch the perpendicular/diagonal split and label both halves before writing any equation.
Treating a “close” converse check (e.g. within ) as proof of a right angle.	Reinforce that the converse of Pythagoras’ theorem requires exact equality of squares.
Leaving an answer as rather than taking the final square root.	Make “take the square root” a non-negotiable last line of every solution.
Not sketching a diagram for a worded real-world problem before starting.	Require a labelled sketch as the first step of every applied problem, with the hypotenuse marked clearly.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A cube has a space diagonal of cm. Find its side length.

Answer

Since the space diagonal of a cube of side is , we have , so cm.

E2 (Kangaroo style). Two squares have side lengths cm and cm. A third square’s area equals the sum of the first two squares’ areas. Find the side length of the third square, and identify the resulting Pythagorean triple.

Answer

, so the third square has side cm. The triple is .

E3 (Challenge). A triangle has sides , , . For which value of does this triangle have a perimeter of ?

Answer

Perimeter . Setting gives . Testing whole numbers: gives (too small); gives ; neither gives exactly , so there is no whole-number solution — a good discussion point about when “neat” formulas don’t produce neat answers.

Homework

1. Find the hypotenuse of a triangle with legs cm and cm.
2. Find the missing leg of a triangle with hypotenuse cm and one leg cm.
3. An isosceles triangle has base cm and equal sides cm. Find its height.
4. A rectangular prism measures cm cm cm. Find its space diagonal, to decimal place.
5. Test whether , , and , , are both right-angled triangles. What do you notice?
6. Reasoning. Explain, in your own words, the difference between using Pythagoras’ theorem to find a missing side and using its converse to test for a right angle.
7. Challenge. A ship sails km east then turns and sails km on a bearing that takes it directly toward its starting point’s north-south line, ending up exactly north of its starting point. How far north of the start is the ship, and how far is it from the start in total straight-line distance? (Hint: sketch the right-angled triangle formed.)

Answers: Q1 — cm. Q2 — cm. Q3 — half-base , cm. Q4 — cm. Q5 — : , right-angled; : , not right-angled — a small change to the longest side is enough to break the right angle. Q6 — finding a side assumes the triangle is already right-angled and calculates an unknown length; the converse instead starts from three known lengths and tests whether a right angle exists at all. Q7 — the “10 km leg” is the hypotenuse of the final short right-angled triangle only if reinterpreted; treating the direct distance from start as the hypotenuse of the whole journey with legs km and the northward distance: if the final straight-line distance from start is required to make a right angle with the km east leg, then northward distance for the resulting hypotenuse — this question is intentionally open-ended to prompt discussion of what “directly toward its starting point’s north-south line” means; accept any answer supported by a correctly labelled diagram and consistent working.