The diagram below shows a circle with centre O. The five points A, B, C, D, and E are on the circle. [AB] and [DE] are diameters of the circle, and $ riangle{BOE} = 40^{ ext{o}}$. The angles X, Y, and Z are labelled. (a) Write down the size of the angle X. (b) Work out the size of the angle Y. (c) Work out the size of the angle Z.

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The diagram below shows a circle with centre O - Junior Cycle Mathematics - Question 1 - 2018

Question 1

The diagram below shows a circle with centre O. The five points A, B, C, D, and E are on the circle. [AB] and [DE] are diameters of the circle, and $ riangle{BOE} = ... show full transcript

Worked Solution & Example Answer:The diagram below shows a circle with centre O - Junior Cycle Mathematics - Question 1 - 2018

Step 1

Write down the size of the angle X.

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Answer

The angle X, being subtended by diameter [AB], is equal to the angle subtended at the circumference by the same arc. Therefore, the size of angle X is:

$X = 40^{ ext{o}}$

Step 2

Work out the size of the angle Y.

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Answer

Triangle AOD is isosceles, as OD = OA (radii of the circle). Therefore, we can use the exterior angle theorem:

The angle at AOD is $180^{ ext{o}} - 40^{ ext{o}} = 140^{ ext{o}}$ .
As triangle AOD is isosceles, $2Y = 140^{ ext{o}}$ .
Solving gives:

$Y = \frac{140^{ ext{o}}}{2} = 70^{ ext{o}}$

Step 3

Work out the size of the angle Z.

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Answer

Using the properties of a cyclic quadrilateral, we know:

The opposite angles sum to $180^{ ext{o}}$ . Thus:

$Y + Z = 180^{ ext{o}}$

Substituting for Y:

$70^{ ext{o}} + Z = 180^{ ext{o}}$ $Z = 180^{ ext{o}} - 70^{ ext{o}} = 110^{ ext{o}}$

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