GCSE OCR Maths Angle Facts: An equilateral triangle, a regul

An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point - OCR - GCSE Maths - Question 10 - 2021 - Paper 1

Question 10

An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point. (a) Show that angle A is 156°. (b) Work out the number of sides of... show full transcript

Worked Solution & Example Answer:An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point - OCR - GCSE Maths - Question 10 - 2021 - Paper 1

Step 1

Show that angle A is 156°

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Answer

To find angle A, we first need to determine the internal angles of the polygons involved.

Internal angle of a regular 10-sided polygon: The formula for the internal angle of a regular polygon with n sides is: $ext{Internal Angle} = \frac{(n - 2) \times 180}{n}$ For a 10-sided polygon (n = 10): $\text{Internal Angle} = \frac{(10 - 2) \times 180}{10} = \frac{8 \times 180}{10} = 144°$
Internal angle of an equilateral triangle: An equilateral triangle has all angles equal to 60°.
Calculating angle A: The sum of the angles at point A is equal to 360°: $\text{Angle A} + \text{Angle of 10-sided polygon} + \text{Angle of equilateral triangle} = 360°$ Substituting in the values: $\text{Angle A} + 144° + 60° = 360°$ Therefore: $\text{Angle A} = 360° - 144° - 60° = 156°$

Hence, we have shown that angle A is indeed 156°.

Step 2

Work out the number of sides of the other regular polygon.

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Answer

Let the number of sides of the other regular polygon be n. The internal angle for this polygon can be expressed as: $\text{Internal Angle} = \frac{(n - 2) \times 180}{n}$

From the first part, we have established that:

Angle A = 156°.
The angle at the vertex shared with the 10-sided polygon is 144°.

Thus, at point A, the angle corresponding to the other polygon can be calculated as follows: $\text{Angle} = 360° - 156° - 144° = 60°$

Now, setting the internal angle equal to 60° gives: $\frac{(n - 2) \times 180}{n} = 60$

Cross-multiplying, we have: $(n - 2) \times 180 = 60n$ $180n - 360 = 60n$ $120n = 360$ $n = 3$

Therefore, the other regular polygon is a triangle, which confirms that it has 3 sides.

An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point - OCR - GCSE Maths - Question 10 - 2021 - Paper 1

Worked Solution & Example Answer:An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point - OCR - GCSE Maths - Question 10 - 2021 - Paper 1

Show that angle A is 156°

Work out the number of sides of the other regular polygon.

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