Math / Geometry

QuestionA, B, C and D are four points on the circumference of a circle. TA is the tangent to the circle at A. Angle DAT $=30^{\circ}$ . Angle $\mathrm{ADC}=132^{\circ}$ . a) Calculate the size of angle ABC . Explain your method: b) Calculate the size of angle CBD. Explain your method. c) Explain why AC cannot be a diameter of the circle.

Studdy Solution

STEP 1

1. Points A, B, C, and D lie on the circumference of the circle.
2. TA is a tangent to the circle at point A.
3. Angle DAT is given as $30^\circ$ .
4. Angle ADC is given as $132^\circ$ .
5. We will use properties of circles, tangents, and cyclic quadrilaterals.

STEP 2

1. Use the tangent-segment theorem to find angle BAC.
2. Use the properties of cyclic quadrilaterals to find angle ABC.
3. Use the properties of cyclic quadrilaterals to find angle CBD.
4. Explain why AC cannot be a diameter.

STEP 3

According to the tangent-segment theorem, the angle between the tangent and the chord (angle DAT) is equal to the angle in the alternate segment (angle BAC). Therefore:
$\angle BAC = \angle DAT = 30^\circ$

STEP 4

Since A, B, C, and D are points on the circumference, they form a cyclic quadrilateral. The opposite angles of a cyclic quadrilateral sum to $180^\circ$ . Therefore:
$\angle ABC + \angle ADC = 180^\circ$
Given $\angle ADC = 132^\circ$ , we can find $\angle ABC$ :
$\angle ABC = 180^\circ - 132^\circ = 48^\circ$

STEP 5

To find $\angle CBD$ , we use the fact that the angles subtended by the same arc in a circle are equal. Since $\angle BAC = 30^\circ$ , and $\angle BAC$ and $\angle CBD$ subtend the same arc BC, we have:
$\angle CBD = \angle BAC = 30^\circ$

STEP 6

For AC to be a diameter, $\angle ABC$ would have to be $90^\circ$ (since the angle subtended by a diameter is a right angle). However, we calculated $\angle ABC = 48^\circ$ , which is not $90^\circ$ . Hence, AC cannot be a diameter.
The solutions are: a) $\angle ABC = 48^\circ$ b) $\angle CBD = 30^\circ$ c) AC cannot be a diameter because $\angle ABC$ is not $90^\circ$ .

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