Mathematics • Year 9 • Unit 3 • Lesson 5
Sides of a Right-Angled Triangle
Build fluency with the three new side names used in trigonometry — hypotenuse (always opposite the 90°), opposite (across from the reference angle $\theta$) and adjacent (next to $\theta$, but not the hypotenuse). Practise the three-step labelling: hyp → opp → adj. Understand why opp and adj SWAP when $\theta$ moves to the other vertex.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. In right triangle $ABC$ the right angle is at $B$. The sides are $AB = 3$, $BC = 4$, $AC = 5$. Mark $\theta$ at vertex $A$. Identify the hypotenuse, the opposite, and the adjacent.
Step 1 — Find the hypotenuse first.
The right angle is at $B$. The side OPPOSITE $B$ (the one not touching $B$) is $AC = 5$. So hyp = $AC = 5$.
Reason: hyp is always opposite the 90° — never the reference angle. Lock this down before anything else.
Step 2 — Find the opposite.
$\theta$ is at $A$. The side across from $A$ (the one not touching $A$) is $BC = 4$. So opp = $BC = 4$.
Reason: opposite means "the side across from $\theta$" — the side you'd see if you stood at $\theta$ and looked across.
Step 3 — Find the adjacent (the leftover).
Remaining side is $AB = 3$ (next to $\theta$ at $A$, but not the hypotenuse). So adj = $AB = 3$.
Reason: adjacent is the LEG that touches $\theta$ — never the hypotenuse, which is a category on its own.
Answer: hyp = $AC = \mathbf{5}$; opp = $BC = \mathbf{4}$; adj = $AB = \mathbf{3}$.
2. We do — fill in the missing steps
Same triangle as Section 1 ($AB = 3$, $BC = 4$, $AC = 5$, right angle at $B$), but now $\theta$ has MOVED to vertex $C$. Re-label the sides. 4 marks
Problem. Triangle $ABC$, right angle at $B$, $AB = 3$, $BC = 4$, $AC = 5$. $\theta$ is at vertex $C$. Identify hyp, opp, adj.
Step 1 — Hyp (unchanged): the right angle is still at $B$, so hyp is still the side opposite $B$, which is __________ = ____.
Step 2 — New opposite: $\theta$ is at $C$, so opp is the side across from $C$ (the one NOT touching $C$). That side is __________ = ____.
Step 3 — New adjacent: the remaining leg, next to $\theta$ at $C$ but not the hyp, is __________ = ____.
Compared with Section 1: hyp $\_\_\_\_\_\_\_\_$ (same / changed); opp and adj $\_\_\_\_\_\_\_\_$ (same / swapped).
3. You do — independent practice
Show your reasoning under each problem. The first four are foundation (label one side at a time). The middle two are standard (label all three sides). The last two are extension (the swap, and a tricky orientation).
Foundation — one side at a time
3.1 In a right triangle, which side is the hypotenuse — relative to the right angle? (One short sentence.) 1 mark
3.2 In a right triangle with $\theta$ at vertex $A$, which side is the "opposite" relative to $\theta$? (One short sentence.) 1 mark
3.3 In a right triangle with $\theta$ at vertex $A$, which side is the "adjacent" relative to $\theta$? (One short sentence — make sure to exclude the hypotenuse.) 1 mark
3.4 True or false: the hypotenuse stays in the same place even when $\theta$ moves to a different vertex. Justify in one sentence. 1 mark
Standard — label all three sides
3.5 In right triangle $PQR$ the right angle is at $Q$. $PQ = 6$, $QR = 8$, $PR = 10$. Mark $\theta$ at $P$. Identify hyp, opp and adj (with both name and length). 2 marks
3.6 Same triangle as 3.5 ($PQR$, right angle at $Q$, $PQ = 6$, $QR = 8$, $PR = 10$). Now $\theta$ is at $R$. Identify hyp, opp and adj. State which of opp and adj have swapped compared with 3.5. 2 marks
Extension — tricky orientations
3.7 A right triangle has its right angle marked, but the triangle has been rotated so that the hypotenuse runs vertically. Sketch this and state where the hypotenuse, opposite, and adjacent are (with $\theta$ at one of the acute angles of your choice). 3 marks
3.8 A classmate says "the opposite is always the vertical side." Give one specific example (with $\theta$'s position stated) where this is false. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do ($\theta$ at $C$)
Step 1: hyp = $\mathbf{AC} = \mathbf{5}$ (still opposite the right angle at $B$).
Step 2: opp = $\mathbf{AB} = \mathbf{3}$ (the side across from $C$ — not touching $C$).
Step 3: adj = $\mathbf{BC} = \mathbf{4}$ (the remaining leg, next to $\theta$ at $C$ but not hyp).
Compared with Section 1: hyp $\mathbf{same}$; opp and adj $\mathbf{swapped}$.
3.1 — Hypotenuse
The hypotenuse is the side $\mathbf{opposite\ the\ right\ angle}$ — also the longest side of the triangle.
3.2 — Opposite (relative to $\theta$)
The opposite is the side $\mathbf{across\ from\ \theta}$ — the side that does NOT touch the vertex where $\theta$ is marked.
3.3 — Adjacent (relative to $\theta$)
The adjacent is the $\mathbf{leg\ next\ to\ \theta\ that\ is\ NOT\ the\ hypotenuse}$. It's the leftover — touches $\theta$ but isn't the hyp.
3.4 — Does the hypotenuse move?
$\mathbf{True}$. The hypotenuse is always the side opposite the right angle, which never moves — only $\theta$ moves between the two acute angles. Hyp is fixed; opp and adj swap.
3.5 — Triangle $PQR$, right angle at $Q$, $\theta$ at $P$
hyp = $PR$ (opposite the right angle at $Q$) $= \mathbf{10}$.
opp = $QR$ (across from $P$, not touching $P$) $= \mathbf{8}$.
adj = $PQ$ (next to $\theta$ at $P$, but not hyp) $= \mathbf{6}$.
3.6 — Same triangle, $\theta$ now at $R$
hyp = $PR$ (unchanged — still opposite right angle at $Q$) $= \mathbf{10}$.
opp = $PQ$ (across from $R$, not touching $R$) $= \mathbf{6}$.
adj = $QR$ (next to $\theta$ at $R$, but not hyp) $= \mathbf{8}$.
Compared with 3.5: opp and adj have $\mathbf{swapped}$ (was 8 and 6 — now 6 and 8). Hyp unchanged.
3.7 — Rotated triangle (sample sketch + labels)
Sketch a right triangle with the hypotenuse running vertically (the right angle then sits at the bottom or top corner, with one leg horizontal). Example: right angle at bottom-left, vertical leg going up, horizontal leg going right, hypotenuse running from top of vertical leg down to right end of horizontal leg.
With $\theta$ at the TOP of the hypotenuse (the upper acute angle):
hyp = the vertical (or near-vertical) side opposite the right angle.
opp = the horizontal leg (across from $\theta$).
adj = the vertical leg meeting $\theta$ but NOT the hypotenuse — the leg with the right angle at its base.
Key point: the labels depend on where the right angle and $\theta$ are, not on which side looks "vertical".
3.8 — Opposite is NOT always vertical (counterexample)
Take a standard right triangle in the usual orientation (right angle at bottom-right, horizontal leg going left, vertical leg going up, hypotenuse running diagonally).
Put $\theta$ at the $\mathbf{TOP}$ acute angle (top of the vertical leg). Then the side across from $\theta$ is the $\mathbf{HORIZONTAL\ leg}$ — opposite is now horizontal, not vertical. ✗ Counterexample to the claim.
Conclusion: opp depends on where $\theta$ is, not on which leg looks vertical.