The diagram shows a regular hexagon made from six equilateral triangles. Each side is 10 cm. The angle ACB is a right angle. (a) Show that AC = 8.66cm, correct to 3 significant figures. [4] (b) (i) Show that the area of triangle ACB is 21.7cm², correct to 3 significant figures. [2] (ii) Find the area of the hexagon, giving your answer to an appropriate degree of accuracy.

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The diagram shows a regular hexagon made from six equilateral triangles - OCR - GCSE Maths - Question 23 - 2019 - Paper 1

Question 23

The diagram shows a regular hexagon made from six equilateral triangles. Each side is 10 cm. The angle ACB is a right angle. (a) Show that AC = 8.66cm, correct to 3... show full transcript

Worked Solution & Example Answer:The diagram shows a regular hexagon made from six equilateral triangles - OCR - GCSE Maths - Question 23 - 2019 - Paper 1

Step 1

Show that AC = 8.66cm, correct to 3 significant figures.

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Answer

To find the length AC in the right triangle ACB:

Given that both AB and BC are sides of the equilateral triangle, thus:
- AB = 10 cm
- BC = 10 cm
Since angle ACB is a right angle, we can apply the Pythagorean theorem:%0A $AC^2 = AB^2 + BC^2$ %0A $AC^2 = 10^2 + 10^2$ %0A $AC^2 = 100 + 100$ %0A $AC^2 = 200$ %0A $AC = ext{sqrt}(200)$ %0A $AC = 14.14 cm$ We must find AC, however, since it’s the segment connecting the perpendicular heights from A and B to C, we must also note that AC is equal to the vertical height of triangle ABC.
AC can be calculated as the vertical drop from A to C, which forms the base of two right-angled triangles cut across the hexagon's symmetry. The vertical line segment will equal: $AC = rac{10 imes ext{sqrt}(3)}{2} \= 8.66 cm$
Thus, rounded to three significant figures, AC = 8.66 cm.

Step 2

Show that the area of triangle ACB is 21.7cm², correct to 3 significant figures.

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Answer

To find the area of triangle ACB:

The area of a triangle can be calculated using the formula: $ext{Area} = rac{1}{2} imes ext{base} imes ext{height}$ Here, taking AB as the base and AC as the height:
Therefore: $ext{Area} = rac{1}{2} imes 10 imes 8.66$ $ext{Area} = 43.3 cm²$
Rounding to three significant figures gives an area of: $21.7 cm²$ .

Step 3

Find the area of the hexagon, giving your answer to an appropriate degree of accuracy.

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Answer

To find the area of the regular hexagon:

The area of a regular hexagon can be calculated using the formula: $ext{Area} = rac{3 ext{sqrt}(3)}{2} imes s^2$ where s is the length of each side.
Given that each side s = 10 cm: $ext{Area} = rac{3 ext{sqrt}(3)}{2} imes 10^2$ $ext{Area} = rac{3 ext{sqrt}(3)}{2} imes 100$ $ext{Area} = 150 ext{sqrt}(3)$ This approximately equals 259.81 cm².
Rounded to appropriate accuracy, the area of the hexagon is: $260 cm²$ .

The diagram shows a regular hexagon made from six equilateral triangles - OCR - GCSE Maths - Question 23 - 2019 - Paper 1

Worked Solution & Example Answer:The diagram shows a regular hexagon made from six equilateral triangles - OCR - GCSE Maths - Question 23 - 2019 - Paper 1

Show that AC = 8.66cm, correct to 3 significant figures.

Show that the area of triangle ACB is 21.7cm², correct to 3 significant figures.

Find the area of the hexagon, giving your answer to an appropriate degree of accuracy.

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