Question

QUESTION 10
10.1 In the diagram, 𝐵𝐷 is the diameter of circle 𝐴𝐵𝐶𝐷 with centre 𝑀.
𝐴𝑀𝐶 = 128° and 𝐷
1 = 32°
Figure 10.1: Circle 𝐴𝐵𝐶𝐷 with centre 𝑀.
10.1.1 Calculate, with reasons, the numerical values of 𝐷 .
2
(3)
10.1.2 Calculate, with reasons, the numerical values of 𝐴 .
1+2
(5)
13
10.2 In the diagram, 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral. Chords 𝐴𝐶 and 𝐵𝐷 intersect at 𝐹.
𝐴𝐷 produced meets 𝐵𝐶 produced at 𝐺. 𝐸𝐵 is a tangent to the circle at 𝐵.
𝐴𝐶 produced cuts the tangent 𝐸𝐵 at 𝐸. 𝐴𝐹 = 𝐵𝐹, 𝐴 and .
1 = 𝑦 𝐴
2 = 𝑥
Figure 10.2: Circle 𝐴𝐵𝐶𝐷.
Express, with reasons, the size of the following in terms of 𝑥 and 𝑦:
10.2.1 𝐵
1 (2)
10.2.2 𝐵
2 (2)
10.2.3 𝐷
1 (5)
[17]

Answer 1

To solve questions 10.1 and 10.2 based on the provided geometry and angles related to circles and cyclic quadrilaterals, let’s break down the problem into parts.

### Question 10.1

#### 10.1.1 Calculate, with reasons, the numerical values of \(D_1\).

Since \(BD\) is the diameter of the circle, angle \(BMD\) is \(90^\circ\) (angle in a semicircle).

- Given: \(AMC = 128^\circ\).

In triangle \(AMD\):

\[
\text{Angle } D = 180^\circ - AMC - BMD
\]
\[
D_1 = 180^\circ - 128^\circ - 90^\circ
\]
\[
D_1 = 180^\circ - 218^\circ = -38^\circ.
\]

Since angles cannot be negative, we must reassess \(D_1\).

We are actually calculating angle \(D_1\):

Using the cyclic nature and properties, we also know:
\[
D_1 + AMC = 180^\circ \quad (\text{angles opposite in a cyclic quadrilateral})
\]
\[
D_1 + 128^\circ = 180^\circ.
\]
\[
D_1 = 180^\circ - 128^\circ = 52^\circ.
\]

Thus:
\[
\boxed{D_1 = 52^\circ}.
\]

#### 10.1.2 Calculate, with reasons, the numerical values of \(A_1\).

Using the same cyclic quadrilateral properties:

\[
A_1 + B_1 = 180^\circ,
\]
but we know:

\[
B_1 = D_1 = 52^\circ \quad (\text{as per previous calculation}).
\]

Now we have to calculate \(A_1\):
\[
A_1 = 180^\circ - B_1.
\]
Next, we observe angle \(B\):
\[
A_1 = 180^\circ - 52^\circ = 128^\circ.
\]

Thus:
\[
\boxed{A_1 = 128^\circ}.
\]

### Question 10.2

Using the properties of cyclic quadrilaterals and tangents:

#### 10.2.1 Express \(B_1\) in terms of \(x\) and \(y\).

Since \(E\) is a point where the tangent meets the extended line \(AC\):

By the tangent-secant theorem:
\[
B_1 = y \quad \text{(angle between tangent and chord)}.
\]

Hence,
\[
\boxed{B_1 = y}.
\]

#### 10.2.2 Express \(B_2\) in terms of \(x\) and \(y\).

By the same rationale applied in \(B_2\):
- This angle is subtended by arc \(A\) opposite to it on the opposite side:
\[
B_2 = x \quad \text{(from angle subtending opposite)}.
\]

Thus:
\[
\boxed{B_2 = x}.
\]

#### 10.2.3 Express \(D_1\) in terms of \(x\) and \(y\).

In the case of \(D_1\), we apply the property of angles in opposite arcs:
\[
D_1 + y + x = 180^\circ,
\]
therefore:

\[
D_1 = 180^\circ - (x + y).
\]

Thus,
\[
\boxed{D_1 = 180^\circ - x - y}.
\]

### Summary of Results:
- \(D_1 = 52^\circ\)
- \(A_1 = 128^\circ\)
- \(B_1 = y\)
- \(B_2 = x\)
- \(D_1 = 180^\circ - x - y\)