071i - Composite Figures with Pythagoras' Theorem

Lesson 71 — Applying Pythagoras’ Theorem to Composite Figures

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

Learning Intentions

• To identify right-angled triangles hidden within composite and three-dimensional figures.
• To apply Pythagoras’ theorem, possibly more than once, to solve for unknown lengths in composite figures.

Success Criteria

I can:

1. Identify a right-angled triangle concealed inside a composite shape.
2. Use Pythagoras’ theorem to find the height of an isosceles triangle by bisecting its base.
3. Use Pythagoras’ theorem to find the length of the diagonal of a rectangle.
4. Extend Pythagoras’ theorem to find the diagonal of a rectangular prism, using two applications of the theorem (or the single combined formula).
5. Combine Pythagoras’ theorem with perimeter and area formulas to solve composite problems.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. State Pythagoras’ theorem in your own words, using the letters , , .
2. A right-angled triangle has legs cm and cm. Find the hypotenuse.
3. A right-angled triangle has hypotenuse cm and one leg cm. Find the other leg.
4. Look at an isosceles triangle (not right-angled) drawn on the board, apex at the top. Where is there a hidden right-angled triangle inside it?

Teacher note: Question 4 is the hook — most students won’t have considered that a perpendicular height line creates two right-angled triangles. Do not resolve it; return to it in Activity 1.

Activities

Activity 1 — Explicit Instruction: the Height of an Isosceles Triangle (12 min)

Draw an isosceles triangle with base cm and equal (slant) sides cm. Drop a perpendicular height from the apex to the base.

I do: “The height line drops exactly to the midpoint of the base, because an isosceles triangle is symmetrical. This splits the base into two equal halves of cm each, and creates two identical right-angled triangles, each with hypotenuse cm and one leg cm.”

We do: Isosceles triangle, base cm, equal sides cm.

You do: Find the height of each isosceles triangle.

1. Base cm, equal sides cm.
2. Base cm, equal sides cm.
3. Base cm, equal sides cm.

(Answers: 1. cm 2. cm 3. cm)

Activity 2 — Explicit Instruction: Diagonals of Rectangles and Prisms (12 min)

I do (2D): A rectangle cm by cm. The diagonal is the hypotenuse of the right-angled triangle formed by two adjacent sides.

I do (3D): Extend to a rectangular prism, length cm, width cm, height cm. This takes two applications of Pythagoras’ theorem: first across the base, then up through the prism.

Note the combined formula this reveals: .

We do: Rectangle diagonal ; prism diagonal .

You do:

1. Rectangle diagonal: cm cm.
2. Rectangle diagonal: cm cm.
3. Prism diagonal: cm cm cm.

(Answers: 1. cm 2. cm 3. cm — notice this one comes out to a whole number!)

Activity 3 — Applied Task: the Garden Shed Cross-section (10 min)

Pairs, then whole-class share.

A garden shed’s front wall, seen side-on, is shaped like a rectangle with a triangular roof on top. The rectangle is m wide and its walls are m tall. The roof is an isosceles triangle, centred on the rectangle, rising a further m above the walls.

Find (a) the total perimeter of the front wall shape (the outside outline only), and (b) its total area.

Socratic scaffolding:

Prompt	Purpose
Understand: what shape is this, really?	A rectangle with an isosceles triangle sitting on top — two familiar shapes joined along one edge.
Which edge is not part of the outer perimeter?	The top of the rectangle (where the roof sits) — it’s an internal line, not an outside edge.
What don’t we know yet?	The length of each sloping roof edge.
Devise a plan	Use Pythagoras on half the roof triangle to find the slant length, then add up all outer edges for perimeter, and add the two shape areas for total area.
Carry out the plan — the roof slant	Half the base is m, the roof height is m, so slant m.
Carry out the plan — perimeter	m (base + 2 walls + 2 slants).
Carry out the plan — area	.
Looking back	Does m sound reasonable for a small shed’s outline? Does look right compared to the rectangle alone (the roof should add a modest amount)?

Checks for Understanding

(6 minutes — exit ticket, collected)

1. An isosceles triangle has base cm and equal sides cm. Find its height.
2. A rectangle measures cm by cm. Find the length of its diagonal.
3. A rectangular prism measures cm cm cm. Find the length of its space diagonal, to decimal place.
4. Explain why you must halve the base of an isosceles triangle before applying Pythagoras’ theorem to find its height.

Answers: 1. Half-base , , cm; 2. , cm; 3. , cm; 4. The height line splits the isosceles triangle into two congruent right-angled triangles, each using only half the base as one leg — using the full base would not form a right angle with the height.

Common Misconceptions

Misconception	How to pre-empt it
Using the full base, not half the base, as the leg in an isosceles triangle problem.	Always sketch the height line first and mark the two equal half-base segments before writing any equation.
Treating the diagonal of a rectangular prism as the same as the diagonal of one of its faces.	Emphasise the two-step process: base diagonal first, then “lift” it through the height in a second right-angled triangle.
Forgetting to take the square root at the final step, leaving the answer as .	Insist the last line of every working always states the answer in the correct units, not the squared value.
Adding side lengths directly instead of squaring them (e.g. instead of ).	Return to the definition: Pythagoras’ theorem relates the squares of the sides, never the sides themselves.
Including an internal line (e.g. the rectangle/triangle join) in a composite perimeter.	Trace the outer boundary with a highlighter before calculating; internal lines are never part of the perimeter.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A square has side length cm. Find the length of its diagonal, correct to decimal place.

Answer

E2 (Kangaroo style). A cube has side length cm. Find the length of the diagonal running from one corner to the opposite corner, correct to decimal place.

Answer

In general, a cube of side has space diagonal .

E3 (Challenge). Squares are drawn on each side of a right-angled triangle with legs cm and cm. Find the total area of all three squares.

Answer

The hypotenuse is cm, so the three squares have areas , and . Total . Notice this is exactly double the largest square’s area (), since means the two smaller squares together already equal the largest.

Homework

1. An isosceles triangle has base cm and equal sides cm. Find its height.
2. Find the diagonal of a rectangle measuring cm by cm.
3. Find the space diagonal of a rectangular prism measuring cm cm cm.
4. A shed front is a rectangle m wide with m walls, topped by a centred isosceles roof rising a further m. Find (a) the total outer perimeter and (b) the total area of the shed front.
5. Reasoning. Explain, using a labelled diagram, why finding the space diagonal of a rectangular prism requires two separate applications of Pythagoras’ theorem rather than one.
6. Challenge. A cube has a space diagonal of length cm. Find the side length of the cube, correct to decimal place.

Answers: Q1 — half-base , , cm. Q2 — , cm. Q3 — , cm. Q4 — roof half-base , roof height , slant m; perimeter m; area . Q5 — the base diagonal and the prism’s height are perpendicular but do not lie in the same plane as either original edge alone, so a first right-angled triangle (across the base) must be solved before a second right-angled triangle (up through the solid) can be formed using that result. Q6 — cm.