Grade 9

The inscribed angle theorem: practice problems

Inscribed angles on the same arc are equal and half the central angle. The key is spotting which angles stand on the same arc.

Basic, Standard, Advanced: Grade 9 · Term 3

BasicStandardAdvancedMix

0Make a worksheet on this topic

The theorem

Inscribed angle theorem

\angle \mathrm{APB} = \frac{1}{2}\angle \mathrm{AOB}

An angle in a semicircle is $90^\circ$ , and in one circle inscribed angles are proportional to their arcs.

Tips

Two radii make an isosceles triangle.
With crossing chords, combine equal inscribed angles with the exterior angle property.
With $n$ equally spaced points, one small arc gives an inscribed angle of $180^\circ \div n$ .

Circles and similarity

If chords AB and CD meet at P, then $\triangle \mathrm{PAC} \sim \triangle \mathrm{PDB}$ , so $\mathrm{PA} \times \mathrm{PB} = \mathrm{PC} \times \mathrm{PD}$ .

Q1BasicGr 9 · Term 3The inscribed angle theorem

In the figure, O is the centre of the circle. Find $\angle x$ .

Answer

\angle x = 58^\circ

Solution

OB = OC, so
$\angle \mathrm{BOC} = 180^\circ - 32^\circ \times 2 = 116^\circ$
The inscribed angle is half the central angle:
$\angle x = 116^\circ \div 2 = 58^\circ$

Q2StandardGr 9 · Term 3The inscribed angle theorem

Points A, B, C, D, E, F, G, H, I divide the circle into 9 equal arcs. Find $\angle x = \angle \mathrm{IAC}$ .

Answer

\angle x = 120^\circ

Solution

Inscribed angle on one small arc:
$360^\circ \div 9 \div 2 = 20^\circ$
$\angle x$ stands on arc CI, which is 6 small arcs:
$\angle x = 20^\circ \times 6 = 120^\circ$

Q3AdvancedGr 9 · Term 3The inscribed angle theorem

Chords AB and CD meet at P. $\mathrm{PA} = 6\text{ cm},\ \mathrm{PB} = 4\text{ cm},\ \mathrm{PC} = 8\text{ cm}$ . Find PD.

Answer

3\text{ cm}

Solution

Inscribed angles on arc BC give $\angle \mathrm{PAC} = \angle \mathrm{PDB}$ , and vertical angles $\angle \mathrm{APC} = \angle \mathrm{DPB}$ , so $\triangle \mathrm{PAC} \sim \triangle \mathrm{PDB}$ .
$\mathrm{PA} : \mathrm{PD} = \mathrm{PC} : \mathrm{PB}$
Substituting,
$6 : x = 8 : 4 \quad\Rightarrow\quad 8x = 24 \quad\Rightarrow\quad x = 3$

Q1BasicGr 9 · Term 3The inscribed angle theorem

In the figure, O is the centre of the circle. Find $\angle x$ .

Q2StandardGr 9 · Term 3The inscribed angle theorem

Points A, B, C, D, E, F, G, H divide the circle into 8 equal arcs. Find $\angle x = \angle \mathrm{CAG}$ .

Q3AdvancedGr 9 · Term 3The inscribed angle theorem

Chords AB and CD meet at P. $\mathrm{PA} = 4\text{ cm},\ \mathrm{PB} = 6\text{ cm},\ \mathrm{PC} = 3\text{ cm}$ . Find PD.

Teleport

Main features

On this page

Topics

Display

The inscribed angle theorem: practice problems

The theorem

Tips

Circles and similarity

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Math problem generator

The theorem

Tips

Circles and similarity

Worked examples

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Practice problems

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Related topics

Parallel lines and polygon angles

Similar figures