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4.8 C) Cyclic Quadrilateral
4.8 C) Cyclic Quadrilateral
A Cyclic quadrilateral is a quadrilateral that is drawn inside a circle. All 4 corners of a cyclic quadrilateral must touch the circumference of the circle. The circle on the left contains a cyclic quadrilateral because all of the corners of the quadrilateral touch the circumference of the circle. The circle on the right does not contain a cyclic quadrilateral and this is because one of the corners of the quadrilateral does not touch the circumference of the circle.
Cyclic Quadrilateral Theorem
The theorem to do with cyclic quadrilaterals is that opposite angles in cyclic quadrilaterals add up to 180°.
The theorem to do with cyclic quadrilaterals is that opposite angles in cyclic quadrilaterals add up to 180°.
The theorem states that opposite angles in a cyclic quadrilateral will add up to 180°. This means that for the cyclic quadrilateral above angle a and angle c will add up to 180°. Also, angle b and angle d will add up to 180°.
Example 1
What is the size of angle x and angle y?
What is the size of angle x and angle y?
The circle theorem states that opposite angles in a cyclic quadrilateral add up to 180°. This means that angle x and the angle that is 104° will add up to 180°. This gives us the following equation:
We want to find the value of x. This means that we need to move the 104 that is currently on the left side of the equation to the right, which we are able to do by doing the opposite; we take 104 from both sides of the equation.
Therefore, angle x is 76°.
Let’s now find the value for y. From the same circle theorem, we know that angle y and the angle that is 63° will add up to 180°. This gives us the following equation:
Let’s now find the value for y. From the same circle theorem, we know that angle y and the angle that is 63° will add up to 180°. This gives us the following equation:
In order to find the value of y, we need to move the 63 from the left to the right, which we can do by taking 63 from both sides of the equation.
Therefore, y is 117°.
Example 2
Find the value of x.
Find the value of x.
There is a cyclic quadrilateral in the above circle because all of the corners touch the circumference of the circle. We know that the rule with a cyclic quadrilateral is that opposite angles in the quadrilateral add up to 180°.
We are given 3 of the angles in the cyclic quadrilateral. Two of these angles are opposite one another and the angles that are opposite one another are the angle that is 7x and the angle that is 3x. These two angles will add up to 180°. We can create the equation below from this information:
We are given 3 of the angles in the cyclic quadrilateral. Two of these angles are opposite one another and the angles that are opposite one another are the angle that is 7x and the angle that is 3x. These two angles will add up to 180°. We can create the equation below from this information:
The first step in solving this equation is to add the two terms on the left together.
We want to find the value of x and not 10x. Therefore, we divide both sides of the equation by 10.
Therefore, the value of x is 18°.
Example 3
What is the size of angle y? Give reasons for your answer.
What is the size of angle y? Give reasons for your answer.
We will need to use two different circle theorems in order to be able to answer this question. We can see from looking at the diagram that we have a cyclic quadrilateral, so we will probably be using the circle theorem whereby opposite angles in a cyclic quadrilateral add up to 180°. However, we do not have any angles for the cyclic quadrilateral, which means that we are unable to use this circle theorem at the moment.
Also, from looking at the diagram we can see that we have a spaceship like shape, and we saw this shape when we looked at the first circle theorem in this whole section. The first circle theorem stated that the angle that is subtended by an arc at the centre of the circle is twice the angle at the circumference of the circle. This means that the angle at the centre (the angle that is 84°) will be twice the size of the angle at the circumference, which is the angle at the top (we can also view this as the angle at the top being half the size of the angle at the centre). This means that the angle at the top will be 42° (84° ÷ 2).
Also, from looking at the diagram we can see that we have a spaceship like shape, and we saw this shape when we looked at the first circle theorem in this whole section. The first circle theorem stated that the angle that is subtended by an arc at the centre of the circle is twice the angle at the circumference of the circle. This means that the angle at the centre (the angle that is 84°) will be twice the size of the angle at the circumference, which is the angle at the top (we can also view this as the angle at the top being half the size of the angle at the centre). This means that the angle at the top will be 42° (84° ÷ 2).
We are now able to use the cyclic quadrilateral circle theorems, which is that the opposite angles in a cyclic quadrilateral add up to 180°. This means that the angle at the top (the one that is 42°) and the angle that we are looking for (angle y) will add up to 180°. This means that we can create the equation:
We find the value of y by moving the 42 from the left to the right, which we do by taking 42 from both sides of the equation.
Therefore, y is 138°.
Final Note
It may be the case that we are given the ratio between two opposing angles in a cyclic quadrilateral. We know from the circle theorem that these two angles will add up to 180°. Therefore, we can view questions like this as “180° is shared in the ratio A:B” and answer the question as if it was a ratio question. There is a question like this in the quiz (question 5).
It may be the case that we are given the ratio between two opposing angles in a cyclic quadrilateral. We know from the circle theorem that these two angles will add up to 180°. Therefore, we can view questions like this as “180° is shared in the ratio A:B” and answer the question as if it was a ratio question. There is a question like this in the quiz (question 5).