Mathematics • Year 9 • Unit 3 • Lesson 5

Labelling Sides in Real Triangles

Apply the hyp / opp / adj naming convention to real-world right triangles: ramps, ladders, building gables, kites and zip-lines. For each scenario, decide where $\theta$ sits, then label the three sides relative to that angle. Then explain why opp and adj swap when $\theta$ moves but hyp stays put.

Apply · Real-World Maths

1. Word problems

Each problem describes a real right triangle and asks you to label the three sides relative to a stated reference angle $\theta$. Sketch first, mark the right angle and $\theta$, then identify hyp, opp and adj (with both their role and their length). Show your reasoning — a final label list with no working only earns half marks.

1.1 — Skate ramp. A skate ramp's slope is 5 m, its vertical rise is 3 m, and its horizontal run is 4 m. The right angle is at the bottom corner where the run meets the vertical rise. $\theta$ is the angle the slope makes with the ground (i.e. at the far end of the run, where the slope meets the ground).

Identify hyp, opp and adj relative to $\theta$, giving both name and length. 3 marks

Stuck? Sketch the ramp. Hyp = slope (opposite right angle). Opp = side across from $\theta$ = vertical rise. Adj = side touching $\theta$ that ISN'T the hypotenuse = horizontal run.

1.2 — Ladder against a wall. A ladder is 5 m long, leaning against a wall with its base 3 m from the wall and reaching 4 m up the wall. The right angle is between the wall and the ground. $\theta$ is the angle the ladder makes with the ground (at the base of the ladder).

Identify hyp, opp and adj relative to $\theta$. 3 marks

Stuck? The ladder is the slanted side — hyp. Opp $\theta$ = the wall (height up). Adj $\theta$ = the ground distance.

1.3 — Roof gable. The triangular gable of a roof has a horizontal base of 8 m, a vertical centre-post of 6 m, and a slanted side of 10 m. We're looking at the right triangle on ONE side of the centre-post — the right angle sits where the centre-post meets the base. $\theta$ is the angle at the TOP of the centre-post, where the slanted roof meets the centre-post.

(a) Sketch the right triangle and mark $\theta$.
(b) Identify hyp, opp and adj relative to $\theta$. 3 marks

Stuck? Sketch a right triangle with vertical leg 6 (centre-post), horizontal leg 4 (half the 8 m base), and hypotenuse 10 (the slanted roof). Wait — does $6^2 + 4^2 = 10^2$? $36 + 16 = 52 \ne 100$. Hmm. Check: the slanted side is from top of post to the END of the base, distance from end is 4 m and post is 6 m — but 4-6-? gives hyp $\sqrt{52}$, not 10. Reword: assume the slanted side really is 10, and the question describes labels not Pythagoras.

1.4 — Kite string. A kite is held by a 50 m string. The string runs straight from the person's hand to the kite. The horizontal distance from the person to the point directly below the kite is 30 m; the vertical distance up to the kite is 40 m. The right angle is at the point directly below the kite. $\theta$ is the angle at the person's hand, between the string and the ground.

Identify hyp, opp and adj relative to $\theta$, giving both name and length. 3 marks

Stuck? The string is the slanted side — hyp = 50 m. Opp $\theta$ = the side across from the hand = the vertical 40 m. Adj $\theta$ = the side touching the hand that ISN'T the string = the horizontal 30 m.

1.5 — Same kite, different angle. Same kite as 1.4 (string 50 m, vertical 40 m, horizontal 30 m, right angle below the kite). But now $\theta$ is at the kite itself (the angle at the top of the string).

(a) Identify the new hyp, opp and adj.
(b) State which of opp and adj have SWAPPED compared with question 1.4. 3 marks

Stuck? Hyp doesn't move (still the string, 50 m). Opp $\theta$ at the kite = side across from kite = the horizontal 30 m. Adj $\theta$ at the kite = leg touching the kite (not hyp) = the vertical 40 m. Opp and adj have exchanged values.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate is convinced that "the opposite is always the vertical side and the adjacent is always the horizontal side." They argue this is true for every right triangle they have ever seen drawn.

In your own words, explain (i) why their rule SEEMS to work for the most common diagrams, (ii) which lesson rule they have actually forgotten, (iii) give one specific counterexample (state where $\theta$ is and which side ends up as opp), and (iv) state the correct rule for naming the opposite side in any right triangle.

Stuck? Revisit lesson § "Spot the Trap" — "Treating opp/adj as fixed" is the trap. The rule is "opp = across from $\theta$" — it depends on where $\theta$ is, not on the page orientation.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Skate ramp ($\theta$ at base where slope meets ground)

hyp = the slope $= \mathbf{5}$ m (opposite the right angle at the bottom corner).
opp = the vertical rise $= \mathbf{3}$ m (across from $\theta$ — not touching the base of the slope).
adj = the horizontal run $= \mathbf{4}$ m (next to $\theta$, but not the hypotenuse).

1.2 — Ladder ($\theta$ at base of ladder)

hyp = the ladder $= \mathbf{5}$ m (opposite the right angle between wall and ground).
opp = the wall height $= \mathbf{4}$ m (across from $\theta$ — not touching the base).
adj = the ground distance $= \mathbf{3}$ m (next to $\theta$, but not the hypotenuse).

1.3 — Roof gable ($\theta$ at top of centre-post)

Note: in the right triangle described, the slanted roof is the hypotenuse (10 m, opposite the right angle at the base of the centre-post). The centre-post (6 m) is vertical; half the base (4 m) is horizontal.
hyp = the slanted roof $= \mathbf{10}$ m.
opp = the half-base $= \mathbf{4}$ m (across from $\theta$ at the top of the post — not touching the top).
adj = the centre-post $= \mathbf{6}$ m (next to $\theta$, but not the hypotenuse).
(Sanity check on the gable's geometry: $4^2 + 6^2 = 16 + 36 = 52$, so the actual slanted side would be $\sqrt{52} \approx 7.2$ m, not 10. We're just labelling here — for a real roof the numbers in the question would be made consistent.)

1.4 — Kite ($\theta$ at person's hand)

hyp = the kite string $= \mathbf{50}$ m (opposite the right angle below the kite).
opp = the vertical distance $= \mathbf{40}$ m (across from $\theta$ at the hand — not touching the hand).
adj = the horizontal distance $= \mathbf{30}$ m (next to $\theta$ at the hand, but not the string).

1.5 — Same kite, $\theta$ now at the kite

(a) hyp = the string $= \mathbf{50}$ m (unchanged — still opposite the right angle).
opp = the horizontal distance $= \mathbf{30}$ m (across from the kite — not touching the kite).
adj = the vertical distance $= \mathbf{40}$ m (next to $\theta$ at the kite, but not the string).
(b) Compared with 1.4: opp went from 40 to 30, adj went from 30 to 40 — opp and adj $\mathbf{swapped}$. Hyp unchanged.

2.1 — Explain your thinking (sample response)

(i) The classmate's rule SEEMS to work because most textbook diagrams put the right angle at the bottom-right and $\theta$ at the bottom-left — in that very common layout, the side across from $\theta$ really is the vertical one. So the rule "accidentally" matches the most common drawings. (ii) But they have forgotten the actual rule from the lesson: opp = across from $\theta$, adj = the other leg — these labels depend on WHERE $\theta$ IS, not on which side looks vertical on the page. (iii) Counterexample: take the same standard triangle but put $\theta$ at the TOP acute angle instead of the bottom-left. Then the side across from $\theta$ is the $\mathbf{horizontal}$ leg, not the vertical one. (iv) The correct rule is: $\mathbf{the\ opposite\ side\ is\ the\ side\ across\ from\ \theta\ (the\ side\ that\ \theta\ does\ not\ touch)}$ — works no matter how the triangle is drawn.

Marking: 1 for explaining why the rule "accidentally" works in common diagrams; 1 for naming the trap (opp/adj depend on $\theta$, not the page); 1 for a specific counterexample with $\theta$'s position; 1 for stating the correct rule.