IEO Level 2- English Olympiad (SOF) Class 4 Coaching Programs
⏳ 🎯 Online Tests (3 Tests [35 Questions Each]): NTA Pattern, Analytics & Explanations
Click Here to View & Get Complete Material
Rs. 210.00
3 Year Validity (Multiple Devices)
🎓 Study Material (101 Notes): 2024-2025 Syllabus
Click Here to View & Get Complete Material
Rs. 250.00
3 Year Validity (Multiple Devices)
🎯 240 MCQs (& PYQs) with Full Explanations (2024-2025 Exam)
Click Here to View & Get Complete Material
Rs. 200.00
3 Year Validity (Multiple Devices)
NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Short Answer
Question
In figure, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O. in such a way that the sides AB. BC. CD and DA touch the circle at the points P. Q, R and S respectively. Prove that
Solution
We know that tangents drawn to a circle from an outer points are equal.
So,
CR = CQ and DR = DS.
Now, consider
Hence proved.
Question
If given figure, AP and BP are tangents to a circle with centre O. such that AP = 5 cm and , Find the length of chord AB.
Solution
In we have AP = BP
[ Tangents from an external point arc equally inclined lo segment joining centre to point]
Let .
Then in
As all three angles of APB are . So APH is an equilateral triangle.
Hence
Question
In given figure, a circle is inscribed in a . Such that it touches the sides AB, BC and CA at points D. E and F respectively. If the lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF
Solution
Given.
Let ( Tangent drawn from external point to circle arc equal)
Question
In figure given, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A. If . , then find .
Solution
( Linear Pair Axiom)
Now, (angIe between radius OA and tangent AC is )
Now, in ,
( sum of angles in triangle is )
Question
In given figure. PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and , find
Solution
Construction: Join AO.
Given: PQ is tangent. AB is diameter
To find:
Solution: In ( Equal radii)
( Angles opposite to equal sides are equal)
But, (i)
Since, OC PO ( Tangent is perpendicular to radius at point of contact)
Question
From an external point P, tangents PA and PB are drawn to a circle with centre O. If , then find .
Solution
Given,
( angle between radius OA and tangent PA is )
Now, (∵ Tangents from an external point are same)
(∵ Angle between radius OB and tangent PB is )
Now in AOB we have
PA = PB
( sum of angles in triangle is )
Question
The in circle of an isosceles triangle ABC, with AB = AC, touches the sides AB, BC and CA at D, E and F respectively. Prove that E bisects BC.
Solution
We have - (1)
- (2)
(Tangents drawn from an external point are equal)
On subtracting 2 from 1 we get
- (3)
Also - (4)
From 3 and 4 we get
- (5)
Also, - (6)
From 5 and 6
E bisects BC
Question
In given fig. from an external point P, two tangents PT and PS are drawn to a circle center O and radius r. If PO = 2r, show that
Solution
Let
In right triangle OTP we have
Hence
, wehave OT = OS
In ,
Hence proved
Question
In given fig. a circle inscribed in touches its sides BC, CA and AB at the points P, Q, R respectively. If AB = AC then prove that BP = CP.
Solution
( , equal tangents)
(lengths of equal tangents)
Question
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Solution
Given: A circle with center O.
With tangent XV at point of contact P.
Top rove:
Proof: Let Q be point on XV
Connect OQ
Suppose it touches the circle at R
Hence,
(As OP = OR radius)
Same will be the case with all other points on circle
Hence, OP is the smallest line that connects XV
Question
A circle is inscribed in the quadrilateral ABCD Given BC = 38 cm, BQ = 27 cm. CD = 25 cm find the radius of the circle
Solution
Say the side BC touches the circle at point, the side CD touches the circle at point R and the side DA touches the circle at point S and side AB touches the circle at point P. Also let the center of circle be O:
And hence
, hence
and is a square.
Radius of the circle
Question
Find the value of x.
Solution
Length of tangents from external point are equal.