Right-triangle geometry | Year 11

01 / LEARN

Understand the idea.

The right angle fixes the hypotenuse

Find the marked right angle first. The side opposite it is the hypotenuse, which is the longest side of a right triangle. Its identity stays fixed when the sketch is rotated or another reference angle is chosen.

The remaining two sides are the legs. Since a triangle's angles sum to $180^{\circ}$ , the two acute angles in a right triangle sum to $90^{\circ}$ . Mark known angles before naming the sides.

Opposite and adjacent depend on the angle

Choose one acute reference angle. Its opposite side lies across the triangle and does not touch that vertex. Its adjacent leg touches the reference angle but excludes the hypotenuse, even though the hypotenuse also touches the angle.

Changing to the other acute angle exchanges opposite and adjacent. Write the vertex beside the labels when needed: “opposite A” is unambiguous, while “the vertical side” depends on how the triangle was drawn.

A + C = 90^{\circ} when B = 90^{\circ}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Triangle ABC has B=90° and A=34°. Name the hypotenuse and find C.

Worked solution

1
Locate the side opposite the right angle.
$B = 90^{\circ} \Rightarrow hypotenuse A C$
AC is the only side that does not meet vertex B.
2
Use the acute-angle sum.
$A + C = 90^{\circ}$
Subtracting the right angle from the triangle angle sum leaves ninety degrees.
3
Find the remaining angle.
$C = 90^{\circ} - 34^{\circ} = 56^{\circ}$
The two acute angles are complementary.

Answer

Hypotenuse AC; C=56°.

EXAMPLE 02

Worked example 2

In the same triangle, identify opposite and adjacent legs first for angle A, then for angle C.

Worked solution

1
Keep the hypotenuse fixed.
$hypotenuse A C$
The right angle remains at B for either reference angle.
2
Name the legs relative to A.
$opposite A = B C, adjacent to A = A B$
BC lies across from A; AB touches A and is not the hypotenuse.
3
Name the legs relative to C.
$opposite C = A B, adjacent to C = B C$
AB lies across from C; BC touches C and is not the hypotenuse.

Answer

For A: opposite BC, adjacent AB. For C: opposite AB, adjacent BC.

03 / YOUR TURN

Try it yourself.

Check your understanding

A triangle has angles 90° and 27°. Find its third angle.

Show answer & reasoning

63°

The third angle is $180^{\circ} - 90^{\circ} - 27^{\circ} = 63^{\circ}$ . Equivalently, the two acute angles must add to ninety degrees.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Triangle PQR has $Q = 90^{\circ}$ . Which side is the hypotenuse?
2 marks
In right triangle ABC, $B = 90^{\circ}$ and $A = 41^{\circ}$ . Find C.
2 marks
Triangle XYZ has $Z = 90^{\circ}$ . Relative to angle X, name the opposite leg and adjacent leg.
2 marks
In the previous triangle, choose Y instead. Which leg is now opposite?
2 marks
Could a right triangle have another angle of $94^{\circ}$ ? Explain.
3 marks
A right triangle is drawn upside down. A student calls its shortest side the hypotenuse. Explain the correction.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Triangle PQR has $Q = 90^{\circ}$ . Which side is the hypotenuse?
2 marks
The hypotenuse is opposite the right angle Q.
PR
In right triangle ABC, $B = 90^{\circ}$ and $A = 41^{\circ}$ . Find C.
2 marks
$C = 180^{\circ} - 90^{\circ} - 41^{\circ} = 49^{\circ}$ .
$49^{\circ}$
Triangle XYZ has $Z = 90^{\circ}$ . Relative to angle X, name the opposite leg and adjacent leg.
2 marks
XY is the hypotenuse.
YZ does not touch X; XZ is the remaining leg touching X.
Opposite YZ; adjacent XZ.
In the previous triangle, choose Y instead. Which leg is now opposite?
2 marks
The side not touching Y is XZ; the hypotenuse stays XY.
XZ
Could a right triangle have another angle of $94^{\circ}$ ? Explain.
3 marks
$90^{\circ} + 94^{\circ}$ already exceeds $180^{\circ}$ .
Both non-right angles in a right triangle must be acute.
No.
A right triangle is drawn upside down. A student calls its shortest side the hypotenuse. Explain the correction.
3 marks
Orientation does not change side relationships.
Find the marked right angle, then the opposite side.
The hypotenuse is opposite the $90^{\circ}$ angle and must be the longest side.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.

Official curriculum source ↗

In this lesson

Understand the idea.

The right angle fixes the hypotenuse

Opposite and adjacent depend on the angle

See the method in action.

Worked example 1

Locate the side opposite the right angle.

Use the acute-angle sum.

Find the remaining angle.

Worked example 2

Keep the hypotenuse fixed.

Name the legs relative to A.

Name the legs relative to C.

Try it yourself.

Check your understanding

Keep practising