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Imran joins two tiles together as shown below - OCR - GCSE Maths - Question 8 - 2017 - Paper 1
Question 8
Imran joins two tiles together as shown below. One tile is a regular hexagon and the other tile is a regular pentagon. (a) Show that angle a is 132°. (b) Imran thi... show full transcript
Worked Solution & Example Answer:Imran joins two tiles together as shown below - OCR - GCSE Maths - Question 8 - 2017 - Paper 1
Step 1
Show that angle a is 132°.
Answer
To determine angle a, we first need to calculate the interior angles of the regular hexagon and the regular pentagon.
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Interior angle of a regular hexagon:
The formula for finding the interior angle of a regular polygon is:
For a hexagon, where n = 6:
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Interior angle of a regular pentagon:
For a pentagon, where n = 5:
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Now, calculate angle a:
Since the angle a is supplementary to the interior angle of the pentagon, we can find it as follows:
Therefore:
Simplifying:
Thus, angle a is shown to be 132°.
Step 2
Is Imran correct?
Answer
To determine if another regular polygon can fit exactly into angle a, we need to check the conditions of regular polygons:
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Interior angles of regular polygons must be such that they evenly divide 360°. This means the interior angle must satisfy the equation:
where k is an integer.
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Finding n for angle 132°:
Set :
Simplifying gives:
48n = 360\ n = \frac{360}{48} = 7.5$$ Since n is not an integer, it shows that 132° does not correspond to an angle of a regular polygon. Thus, another tile in the shape of a regular polygon cannot fit exactly into angle a.
Conclusion: Imran is not correct.