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 thinks that another tile in the shape of a regular polygon will fit exactly into angle a. Is Imran correct? Show your reasoning.

GCSE OCR Maths Angle Facts: Imran joins two tiles together a

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°.

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Answer

To determine angle a in the configuration of a regular hexagon and a regular pentagon, we first calculate the interior angles of both shapes.

Interior angle of a regular hexagon: A regular hexagon has 6 sides. The formula for the interior angle of a regular polygon is: $ext{Angle} = \frac{(n - 2) \times 180°}{n}$ where n is the number of sides. Thus, for a hexagon: $\text{Angle} = \frac{(6 - 2) \times 180°}{6} = \frac{720°}{6} = 120°$
Interior angle of a regular pentagon: A regular pentagon has 5 sides. Using the same formula: $\text{Angle} = \frac{(5 - 2) \times 180°}{5} = \frac{540°}{5} = 108°$
Calculating angle a: Since angle a is supplementary to the angles of the hexagon and pentagon:
- The angle a must satisfy: $ext{Angle a} = 360° - (120° + 108°) = 360° - 228° = 132°$

Thus, we have shown that angle a is indeed 132°.

Step 2

Is Imran correct?

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Answer

To determine if another tile in the shape of a regular polygon can fit exactly into angle a, we need to check whether the angle a can be expressed as an interior angle of a regular polygon.

Checking Regular Polygon Angles: The interior angle of a regular polygon can be expressed as: $ext{Interior angle} = \frac{(n - 2) \times 180°}{n}$ Rearranging gives us: $\text{Interior angle} \times n = (n - 2) \times 180°$ Simplifying this: $180n - 360 = 132n$ Thus, $48n = 360; \quad n = \frac{360}{48} = 7.5$
Conclusion: Since n must be an integer (as it represents the number of sides in a polygon), angle a of 132° cannot fit exactly into a regular polygon. Therefore, Imran is incorrect.

Imran joins two tiles together as shown below - OCR - GCSE Maths - Question 8 - 2017 - Paper 1

Worked Solution & Example Answer:Imran joins two tiles together as shown below - OCR - GCSE Maths - Question 8 - 2017 - Paper 1

Show that angle a is 132°.

Is Imran correct?

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