Question
In a circle with centre O , chords AB and CD intersets inside the circle at E . Prove that ∠ AOC = ∠ BOD = 2 ∠ AEC.
✓
Answer
Arc AC subtends LAOC at the centre of circle and LABC on the circumference of the cirde .
∴ ∠ AOC = 2 ∠ ABC ...(1)
Similarly, ∠ BOD and ∠ DCB are the angles subtended by the arc DB at the centre and on the circumference of the circle respectively .
∴ ∠ BOD = 2 ∠ DCB ... (2)
Adding ( 1) and (2),
∠ AOC+ ∠ BOD = 2(∠ ABC + ∠ DCB) ... (3)
In triangle ECB ,
∠ AEC = ∠ ECB + ∠ EBC = ∠ DCB + ∠ ABC
From (3),
∠ AOC+ ∠ BOD = 2 ∠ AEC
Hence Proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the given figure, I is the incentre of ΔABC. BI when produced meets the circumcircle of
ΔABC at D. ∠BAC = 55° and ∠ACB = 65°; calculate:
(i) ∠DCA, (ii) ∠DAC, (iii) ∠DCI, (iv) AIC
ΔABC at D. ∠BAC = 55° and ∠ACB = 65°; calculate:
(i) ∠DCA, (ii) ∠DAC, (iii) ∠DCI, (iv) AIC
Without solving the following quadratic equation, find the value of 'm' for which the given equation has real and equal roots.
$x^2+ 2 (m - 1) x + (m + 5) = 0.$
→$x^2+ 2 (m - 1) x + (m + 5) = 0.$
Two numbers are in the ratio $5 : 7$ and the difference of their squares is $600$. Find the numbers.
→Given: ∠CAB = 75° and ∠CBA = 50°. Find the value of ∠DAB + ∠ABD.
The ordinate of a point lying on the line joining the points $(6, 4)$ and $(7, -5)$ is $-23$. Find the coordinates
of that point.
→of that point.
The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:
(i) angle QTR
(i) angle QTR
Draw a histogram for the following distribution and estimate the mode:
| Marks% | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 | 90-99 |
| No. of students | 14 | 26 | 40 | 92 | 114 | 78 | 36 |
Mrs Khandelkar invests Rs 900 every month in a recurring deposit account for a period of 3 years at a simple interest rate of 8% pa
(a) Find the total interest she will earn at the end of the period.
(b) Find the maturity value of her deposits.
(a) Find the total interest she will earn at the end of the period.
(b) Find the maturity value of her deposits.
Take a graph paper and mark the points P(2, 1), Q(7, 1) and R(7,5). Taking QR as the line of symmetry, obtain and write the co-ordinates of point S.
Manoj saves $Rs.5,000$ every year and invests it at $12\%$ p.a. compound interest. Calculate his savings at the end of the third year.
→