08. Circle Theorems - Cardiff Tutor Company

Question 1

The points C, B, and D lie on a circle with centre A. $ACD=20^\circ$ and $ABC=40^\circ$ . The straight line passing through D is tangent to the circle. Calculate the value of the angle labelled $x$ .

Select the correct answer from the list below:

A: $25^\circ$

B: $40^\circ$

C: $70^\circ$

D: $85^\circ$

CORRECT ANSWER: C: $70^\circ$

WORKED SOLUTION:

AC and AB are radii, so ABC must be an isosceles triangle, meaning:

ACB=ABC=40^\circ

ABC is a triangle, so the angles must add up to $180^\circ$ .

$ABC+40+40=180$
$ABC+80=180$
$ABC=180-80$
$ABC=100$

We know that the angle at the centre is twice the angel at the circumference, so we can use this to find angle CDB.

$CDB=100\div2$
$CDB=50$

Now, because BCD is a triangle, all the angles have to add up to $180^\circ$ , including the missing angle ABD.

$ABD+40+40+20+50=180$
$ABD+150=180$
$ABD=180-150$
$ABD=180-30$

Using the Alternate Segment Theorem, we can see that $x$ must be the same as angle DBC.

$x=DBC$
$x=30+40$
$x=70$

Question 2

$A, B, C$ , and $D$ lie on a circle with centre $O$ .

Angle $CBD$ is $35^\circ$ .

Find the angle BCD. $x$ .

Select the correct answer from the list below:

A: $65^\circ$

B: $90^\circ$

C: $55^\circ$

D: $35^\circ$

CORRECT ANSWER: C: $55^\circ$

WORKED SOLUTION:

Because $AC$ goes through the centre of the circle, we know that angle $ABC$ must be a right angle.

We can now use this to find the angle $ABD$ .

$ABD+35=90$
$ABD=90-35$
$ABD=55$

Because angles in the same segment are equal, we can see that $x=55$ .

Question 3

The points A, B, C, and D lie one the circle with centre O. Given that BD and AC are perpendicular, and that BC is 7cm, find the length of AB. Explain every step.

Select the correct answer from the list below:

A: $0.7cm$

B: $14cm$

C: $3.5cm$

D: $7cm$

CORRECT ANSWER: D: $7cm$

WORKED SOLUTION:

Because DB is perpendicular to the chord AC, and DB goes through the centre, it means that AC is bisected through a point, say M.

We now have two triangles, CBM and ABM, that have two sides that are the same ( $AC=AM$ and $BM$ ) and the same interior angle ( $90^\circ$ ), meaning they are congruent.

Because these triangles are congruent the final sides must be equal ( $AB=BC=7cm$ ).

Question 4

$A, B,$ and $C$ are points on a circle with centre at $O$ and radius $3cm$ .

$AC$ and $BC$ are tangents to the circle.

$OC$ is 11cm.

Find the area of the shaded region.

Give your answer to $3$ significant figures.