A, B and C are points on a circle of radius 5 cm, centre O - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Question 18
A, B and C are points on a circle of radius 5 cm, centre O.
DA and DC are tangents to the circle.
DO = 9 cm
Work out the length of arc ABC.
Give your answer correct... show full transcript
Worked Solution & Example Answer:A, B and C are points on a circle of radius 5 cm, centre O - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Step 1
Recognize that angle DAB is a right angle
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Since DA and DC are tangents to the circle at points A and C respectively, the angles DAB and DCA are both right angles (90°). Therefore, triangle OAD is a right triangle.
Step 2
Set up an equation using trigonometry for triangle OAD
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the cosine rule, we know that:
extCos(DOA)=DOOA
Given that OA (the radius) is 5 cm and DO is 9 cm, we write:
extCos(DOA)=95
Step 3
Find the angle DOA
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the inverse cosine, we can find the angle:
DOA=Cos−1(95)≈54.74°
Step 4
Calculate the length of arc ABC
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the length of arc ABC, we first determine the angle AOC, which is double the angle DOA:
AOC=2×DOA≈2×54.74°≈109.48°
Next, convert this angle into radians:
AOCrad=180109.48×π≈1.913 rad
Finally, the length of arc ABC can be calculated as:
Length=r×θ=5×1.913≈9.565extcm
Step 5
Round the length to 3 significant figures
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Rounding 9.565 cm to 3 significant figures gives us:
