Mathematics • Year 9 • Unit 4 • Lesson 9
Congruence and Similarity in the Real World
Use the congruence and similarity tests in everyday contexts: a surveyor measuring a river, the shadow of a flagpole, a roof truss made of identical triangles, a kite design, and tessellated patio tiles. Then explain your method in your own words.
1. Word problems
Each problem uses a similarity or congruence test from Lesson 9, with a real-world twist. Show your working, a single final answer with no working only earns half marks.
1.1, Surveying a river. The near bank is the straight line $A$–$B$–$C$, with $AB=30$ m and $BC=40$ m. The tree $T$ is directly across the river from $A$, so $AT$ is perpendicular to the bank. From $C$, the surveyor measures $CD=30$ m perpendicular inland. Standing at $B$, she sights the tree and extends the straight line through $T$ and $B$ inland; it meets the perpendicular through $C$ at $D$. Thus $T,B,D$ are collinear. Use the labelled diagram to find the river width $AT$.
Use the right-angle pair and the equal angles at $B$ to justify similarity, then match corresponding sides to find $AT$. 3 marks
1.2, Flagpole shadow. A flagpole and a $1.6$ m student standing nearby both cast shadows at the same moment. The student's shadow is $2.0$ m long; the flagpole's shadow is $12.5$ m long. The sun's rays hit both at the same angle, so the two right-triangle "person + shadow" and "pole + shadow" diagrams are similar.
(a) State which similarity test applies (the sun's angle is the same for both, and both have a right angle to the ground).
(b) Find the height of the flagpole. 3 marks
1.3, Roof truss. A roof truss is built from $12$ identical right-angled triangles. Each triangle has base $1.2$ m, height $0.9$ m, and hypotenuse $1.5$ m.
(a) Are the triangles congruent or merely similar? Justify with a test.
(b) Find the total length of timber needed for all the hypotenuses combined. 3 marks
1.4, Kite design. A kite has a vertical line of symmetry. The diagonal along this axis divides it into two triangles. Prove that these two triangles are congruent. State the given information, the test you use, and the conclusion.
3 marks
1.5, Tessellated tiles. An outdoor patio is paved with $40$ identical equilateral triangular tiles, side $30$ cm. A landscaper offers a "larger but identical-style" patio with similar tiles of side $45$ cm.
(a) Are the larger tiles congruent or similar to the original tiles? Justify briefly.
(b) What is the scale factor $k$?
(c) By what factor will the total tile area increase? (You may use scale-factor results from Lesson 8.) 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate looks at two equilateral triangles, one with side $5$ cm and one with side $8$ cm, and says "they have all the same angles ($60°$ each), so they must be congruent." In your own words, explain (i) what mistake they have made, (ii) which test they are confusing (and what it actually proves), and (iii) the correct relationship between the two triangles. Refer to "AA" or "AAA" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Surveying a river
Both triangles are right-angled: $\angle TAB=\angle DCB=90°$. At $B$, the angles $\angle TBA$ and $\angle DBC$ are vertically opposite, so they are equal. Therefore $\triangle TAB \sim \triangle DCB$ by AA, with $T\leftrightarrow D$, $A\leftrightarrow C$ and $B\leftrightarrow B$.
Corresponding sides give $\dfrac{AT}{DC}=\dfrac{AB}{CB}$. Thus $\dfrac{AT}{30}=\dfrac{30}{40}$, so $AT=\mathbf{22.5}$ m.
Coordinate witness in metres: $A=(0,0)$, $B=(30,0)$, $C=(70,0)$, $D=(70,-30)$, and $T=(0,22.5)$. These points satisfy both perpendiculars and $T,B,D$ collinear, so both triangles are non-degenerate and the corresponding-side ratio holds.
1.2, Flagpole shadow
(a) The two right triangles have a right angle (ground) and the same sun-angle → similar by AA.
(b) $\dfrac{h}{12.5} = \dfrac{1.6}{2.0} = 0.8$, so $h = 12.5 \times 0.8 = \mathbf{10}$ m.
This is the classic "Thales' theorem" trick, Thales used it 2500 years ago to measure the Great Pyramid.
1.3, Roof truss
(a) All measurements (sides 1.2, 0.9, 1.5) are equal across all 12 triangles → congruent by SSS. Note: $0.9^2 + 1.2^2 = 0.81 + 1.44 = 2.25 = 1.5^2$, confirming right angles. RHS would also work.
(b) Total hypotenuse length $= 12 \times 1.5 = \mathbf{18}$ m of timber.
1.4, Kite
Given: Kite $ABCD$ with line of symmetry $BD$, so $AB = CB$ and $AD = CD$.
To prove: $\triangle ABD \equiv \triangle CBD$.
Proof:
$AB = CB$ (kite, pair of adjacent equal sides)
$AD = CD$ (kite, second pair of adjacent equal sides)
$BD = BD$ (common side)
Therefore $\triangle ABD \equiv \triangle CBD$ by SSS.
Conclusion: The two halves of a kite separated by the symmetry diagonal are congruent triangles.
1.5, Tessellated tiles
(a) The tiles have the same angles ($60°$ all round) but different sides ($30$ vs $45$ cm) → similar, not congruent. Similar by AA (or AAA).
(b) $k = 45/30 = \mathbf{1.5}$.
(c) Each tile's area scales by $k^2 = 2.25$, so total tile area increases by a factor of $\mathbf{2.25}$ (assuming the same number of tiles).
2.1, Explain your thinking (sample response)
My classmate has assumed that equal angles alone are enough to prove congruence, but they only prove similarity, not congruence. The test they are confusing is AA (or AAA): if two (or three) angles of one triangle equal two (or three) angles of another, the triangles are similar, meaning the same shape, but possibly different sizes. The correct relationship is that the two equilateral triangles (side $5$ cm and side $8$ cm) are similar, with scale factor $k = 8/5 = 1.6$. They are not congruent because their sides are different lengths; congruence requires sides to be equal, not merely proportional.
Marking: 1 for naming the mistake; 1 for naming AA/AAA correctly; 1 for the correct relationship (similar, $k = 1.6$); 1 for clear, full-sentence explanation.