QUESTION 10

Question

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:

• \(D_1 = 52^\circ\)
• \(A_1 = 128^\circ\)
• \(B_1 = y\)
• \(B_2 = x\)
• \(D_1 = 180^\circ - x - y\)