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AQA GCSE · Question 23.2 · Geometry and Measures
Here is a cyclic quadrilateral.
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f : g = 2 : 3
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Work out f : h
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Give your answer in its simplest form.
Here is a cyclic quadrilateral.
<br>
f : g = 2 : 3
<br>
Work out f : h
<br>
Give your answer in its simplest form.
How to approach this question
1. Recall the key property of a cyclic quadrilateral concerning its opposite angles.
2. Use this property to set up an equation involving angle f and the 63° angle. Solve it to find the value of f.
3. Use the given ratio f : g = 2 : 3 and your value for f to find the value of g.
4. Use the cyclic quadrilateral property again to set up an equation involving angles g and h. Use your value for g to find h.
5. Now you have values for f and h. Write them as a ratio f : h.
6. Simplify this ratio to its simplest form (with integers).
Full Answer
**1. Find angle f:**
A cyclic quadrilateral is a four-sided shape whose vertices all lie on the circumference of a circle. A property of cyclic quadrilaterals is that opposite angles sum to 180°.
Angle f is opposite the 63° angle.
So, f + 63° = 180°
f = 180° - 63° = 117°.
**2. Find angle g:**
We are given the ratio f : g = 2 : 3.
We know f = 117°, which corresponds to the 2 parts of the ratio.
2 parts = 117°
1 part = 117° / 2 = 58.5°
Angle g corresponds to 3 parts.
g = 3 × 58.5° = 175.5°.
**3. Find angle h:**
Angle g and angle h are opposite angles in the cyclic quadrilateral, so they also sum to 180°.
g + h = 180°
175.5° + h = 180°
h = 180° - 175.5° = 4.5°.
**4. Find the ratio f : h:**
We need to find the ratio f : h in its simplest form.
f : h = 117 : 4.5
To remove the decimal, we can multiply both sides of the ratio by 2.
(117 × 2) : (4.5 × 2)
234 : 9
Now we simplify by dividing both sides by their greatest common divisor. Let's try dividing by 9.
234 ÷ 9 = 26
9 ÷ 9 = 1
The simplified ratio is 26 : 1.
Common mistakes
✗ Forgetting the cyclic quadrilateral theorem.
✗ Making errors when working with the ratio.
✗ Not simplifying the final ratio correctly, or leaving it with a decimal.
✗ Assuming adjacent angles have a special property.
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