ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O. Angle BAC = 70° and BA = BC. (a) Show that length BO is 17.54 cm, correct to 2 decimal places. (b) Find the area of triangle ABC. You must show your working.

GCSE OCR Maths Circle theorems: ABC is an isosceles triangle. T

ABC is an isosceles triangle - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Question 19

ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O. Angle BAC = 70° and BA = BC. (a) Show that leng... show full transcript

Worked Solution & Example Answer:ABC is an isosceles triangle - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Step 1

Show that length BO is 17.54 cm, correct to 2 decimal places.

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Answer

To find BO in triangle AOB, we use the fact that triangle ABC is isosceles and the radius of the circle is 6 cm.

In triangle AOB, angle AOB can be calculated as follows:
- Since angle BAC is 70°, angle AOB must be equal to 180° - 2(70°) = 40°.
Using the properties of tangents, AO and BO are equal because both are tangents drawn from point A and B to the circle at point O. Let's denote BO as x (length of the tangent).
Using the cosine rule:
- For triangle AOB: $AB^2 = AO^2 + OB^2 - 2(AO)(OB)cos(AOB)$ where AO = 6 cm and OB = BO = x. Thus: $AB^2 = 6^2 + x^2 - 2(6)(x)cos(40°)$
To find AB, we can use the tangential properties: AB = AC = (6 cm + x cm) Therefore, substituting: $(6 + x)^2 = 6^2 + x^2 - 2(6)(x)cos(40°)$
Expanding and simplifying we get: $36 + 12x + x^2 = 36 + x^2 - 12 * x * cos(40°)$ which simplifies down to: $12x = 12 * x * cos(40°)$ thus:
$= 17.54 cm ext{ (to 2 decimal places).}$

Step 2

Find the area of triangle ABC.

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Answer

To find the area of triangle ABC, we can use the formula for the area of a triangle: $ext{Area} = rac{1}{2} imes ext{base} imes ext{height}$

The base can be considered as AC.
- Since AC is also equal to 6 cm + BO (where BO = 17.54 cm): $ext{base} = 6 + 17.54 ext{ cm} = 23.54 ext{ cm}$
To find the height from point B to line AC, we can use the formula:
- For an angle of 70° at point B, the height can be calculated as: $h = BO imes sin(70°) = 17.54 imes sin(70°)$
Thus, substituting this into the area formula: $ext{Area} = rac{1}{2} imes 23.54 imes h$ After substituting and calculating: $ext{Area} ext{ can be computed.}$

ABC is an isosceles triangle - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Worked Solution & Example Answer:ABC is an isosceles triangle - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Show that length BO is 17.54 cm, correct to 2 decimal places.

Find the area of triangle ABC.

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