All questions of Circles for Class 10 Exam

Solution:- The length of tangents drawn from an external point to the circle are equal.

In the figure, CA and CB are the tangents from external point of a circle and OA and OB are the two radius of a circle.

Draw a line OC, then you will get two triangles OAC and OBC.

In a ∆le AOC and ∆le BOC,

Angle OAC and OBC = 180 degree. Because, these are angles between the radii and tangents. So, there are two right angles.

The value for right angled triangle is 90 degree. Here, you can see both right angled triangle, so 90×2 = 180 degree.

Angle AOB = Angle OAC + Angle OBC - Angle ACB.

Given:- Angle ACB = 75 degree. , Angle OAC and OBC = 90 degree.

Angle AOB = 90 + 90 - 75.
Angle AOB = 180 -75.
Angle AOB =105 degree.

So, option d is correct friend.

Answer is 12

Given information
- A point P is 25 cm from the centre of a circle.
- The radius of the circle is 7 cm.
- Length of the tangent drawn from P to the circle is x cm.

To find
The value of x.

Solution
Let O be the centre of the circle and PT be the tangent drawn from P to the circle as shown below.

[INSERT IMAGE]

We can observe that OP is the hypotenuse of the right-angled triangle OPT. Therefore, using Pythagoras theorem, we can find the length of PT as follows.

OP² = OT² + PT²
(25)² = (7)² + PT²
625 = 49 + PT²
PT² = 625 - 49
PT² = 576
PT = √576
PT = 24 cm

Therefore, the value of x is 24 cm.

Answer: Option B.

Correct answer is option

Cyclic quadrilaterals do not any definite rules unlike IIgram .The only rule that applies hear is that opp angles are supplementary.Hence it would'n matter if the angle bisectors form a quadrilateral    

C

B

16 cm.

Because that is diameter which definitely pass through the centre

100

Understanding the Circle Geometry
In the given problem, we have a circle with a chord AB, and AOC is its diameter. The angle ACB is given as 50°. We need to find the measure of angle BAT, where AT is the tangent to the circle at point A.
Circle Properties and Tangents
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- The angle between the tangent and the chord through the point of contact is equal to the angle in the alternate segment.
Applying the Theorem
- Since AT is the tangent at point A, and OA is the radius, we have angle OAT = 90°.
- According to the alternate segment theorem, angle BAT (the angle between the tangent AT and chord AB) is equal to angle ACB.
Calculating the Angles
- Given that angle ACB = 50°, we can write:
Angle BAT = Angle ACB
Therefore, Angle BAT = 50°
Conclusion
Thus, the measure of angle BAT is 50°, which corresponds to option 'c'. This is a fundamental property in circle geometry, illustrating the relationship between tangents and chords in a circle.

∠ROQ +∠RPQ = 180°

Theorem use krna if 2 tangents are arising from same point are of equal length

Radius is always perpendicular to the tangent.

AR = AQ (Length of tangents are equal)

In triangle ORQ,

Here, DS = 5 cm

AE = 7 cm (given)

AB IS CORRECT ANS.WER

TRQ=30,
TSR=TRQ=30( alternate segment theorem)-1
TRS=90( angle in semicircle)--2,
from 1&2 and by angle sum property of a triangle,
RTS=60,
since RTS=PRS ( by alternate segment theory)
PRS=60