7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is ... (1) 7.2 In the diagram below, O is the centre of circle CADB. CD is the diameter of the circle. CED ⊥AB with E on AB. AB = 8 units, OC = 5 units, AC = 4/5 units, EO = x units and ∠DBE = 26.6° 7.2.1 Determine, with reasons, the size of each of the following angles: (a) ∠CĨ, (b) ∠A, (c) ∠BĨ. 7.2.2 Write down, without giving reasons, the length of: (a) AE, (b) ED in terms of x. 7.2.3 Hence, or otherwise, determine the numerical value of x.

NSC Technical Mathematics Euclidean Geometry: 7.1 Complete the following theor

7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is .. - NSC Technical Mathematics - Question 7 - 2019 - Paper 2

Question 7

7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is ... (1) 7.2 In the diagram below, O is the centre of c... show full transcript

Worked Solution & Example Answer:7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is .. - NSC Technical Mathematics - Question 7 - 2019 - Paper 2

Step 1

Determine, with reasons, the size of ∠CĨ

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Answer

Given that ∠DBE = 26.6° and that angles in the same segment are equal, we have:

$ext{Let } ext{sin } CĨ = \frac{4}{\frac{4}{5}} = 4.$

From this, since angle ∠CĨ is in the same segment as angle ∠DBE:

∠CĨ = 26.6°.

Step 2

Determine, with reasons, the size of ∠A

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Answer

To find ∠A, we apply the property of angles in a cyclic quadrilateral. We have:

$ext{∠A} = 180° - 90° - 26.6° = 63.4°.$

Thus, ∠A = 63.4°.

Step 3

Determine, with reasons, the size of ∠BĨ

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Answer

For angle ∠BĨ, using the properties of circles:

$ext{∠BĨ} = 90° - 26.6° = 63.4°,$

since it is in a semicircle.

Step 4

Write down, without giving reasons, the length of AE

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Answer

Given that AC = 4/5 units, hence, AE = 4 units.

Step 5

Write down, without giving reasons, the length of ED in terms of x

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Answer

Since AE + ED = AB (8 units), we get: ED = 8 - x.

Step 6

Hence, or otherwise, determine the numerical value of x.

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Answer

Using the derived equation:

$4^2 + (x)^2 = \left(\frac{4}{5}\right)^2$

We find:

x = 3 \, ext{ or } \, x = -13,$$ We take x = 3, since length cannot be negative.

7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is .. - NSC Technical Mathematics - Question 7 - 2019 - Paper 2

Worked Solution & Example Answer:7.1 Complete the following theorem: The line drawn from the centre of a circle to the midpoint of a chord is .. - NSC Technical Mathematics - Question 7 - 2019 - Paper 2

Determine, with reasons, the size of ∠CĨ

Determine, with reasons, the size of ∠A

Determine, with reasons, the size of ∠BĨ

Write down, without giving reasons, the length of AE

Write down, without giving reasons, the length of ED in terms of x

Hence, or otherwise, determine the numerical value of x.

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