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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 lengt... 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.
Answer
To find length BO, we can use the properties of tangents and triangles. In triangle ABC:
- Since angle BAC is 70°, angle OAB will be half this angle (because OB is a radius perpendicular to the tangent at B) making angle OAB = 35°.
- Triangle OAB is thus formed with:
- OA = 6 cm (radius)
- angle OAB = 35°
- AB = AB (we will find this using the law of cosines).
- Using the tangent property:
- By the sine rule in triangle OAB: [ \frac{OB}{\sin(70°)} = \frac{6}{\sin(35°)} ]
- Rearranging gives us: [ OB = 6 \cdot \frac{\sin(70°)}{\sin(35°)} ]
- Calculating this, ( \sin(70°) \approx 0.9397 ) and ( \sin(35°) \approx 0.5736 ), hence:
- [ OB \approx 6 \cdot \frac{0.9397}{0.5736} \approx 9.83\text{ cm} ]
- Therefore, applying the Pythagorean theorem in triangle OAB:
- [ AB^2 = OA^2 + OB^2 ]
- Calculating, [ AB \approx 6^2 + OB^2 ] gives us 17.54 cm as the answer.
Step 2
Find the area of triangle ABC.
Answer
To find the area of triangle ABC:
- We already have the length AB calculated in the previous step (approximately 17.54 cm).
- The height from point O perpendicular to line AC can also be calculated using the radius. Since CO is radius (6 cm):
- Area can be calculated using the formula: [ \text{Area} = \frac{1}{2} \times base \times height ]
- Base AC can be derived considering the isosceles property (where AC = 17.54 cm as well).
- Hence, the area of triangle ABC is: [ \text{Area} = \frac{1}{2} \times AB \times height \approx \frac{1}{2} \times 17.54 \times 6 \approx 52.62 , cm^2 ]