Advertisements
Advertisements
Question
O is the circumcentre of the triangle ABC and D is the mid-point of the base BC. Prove that ∠BOD = ∠A.
Advertisements
Solution
Given: O is the circumcenter of the triangle ABC and D is the midpoint of BC.
To prove: ∠BOD = ∠A
Join OB and OC.
In ΔOBD and ΔCD,
OD = OD ...(Common side)
DB = Dc ...(D is the midpoint of BC)
OB = OC ...(Both are radius of the circle)
By SSS congruence rule, ΔOBD ≅ ΔOCD.
∴ ∠BOD = ∠COD = x (say) ...(By CPCT)
Since, angle subtended by an arc at the center of the circle is twice the angle subtended by it at any other point in the remaining part of the circle, we have:
2∠BAC = ∠BOC
⇒ 2∠BAC = ∠BOD + ∠DOC
⇒ 2∠BAC = x + x
⇒ 2∠BAC = 2x
⇒ ∠BAC = x
⇒ ∠BAC = ∠BOD
Hence proved.
APPEARS IN
RELATED QUESTIONS
In Fig. 1, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB.
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
In following figure, three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these three circles (shaded region). `["Use" pi=22/7]`
If \[d_1 , d_2 ( d_2 > d_1 )\] be the diameters of two concentric circle s and c be the length of a chord of a circle which is tangent to the other circle , prove that\[{d_2}^2 = c^2 + {d_1}^2\].
Find the area of a circle of radius 7 cm.
Draw two circles of different radii. How many points these circles can have in common? What is the maximum number of common points?
Draw a circle of radius of 4.2 cm. Mark its center as O. Takes a point A on the circumference of the circle. Join AO and extend it till it meets point B on the circumference of the circle,
(i) Measure the length of AB.
(ii) Assign a special name to AB.
From the figure, identify two points in the interior.
Say true or false:
Two diameters of a circle will necessarily intersect.
Which is the longest straight line that can be drawn inside a circle?
