Assignment Class X Circles
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ASSIGNMENT
CIRCLES
CLASS X
1. A point P is 26 cm away from the centre O of a circle and the length PQ of the tangent segment drawn from P to the circle is 10 cm. Find the radius of the circle. 2. A circle is touching the side BC of ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ =
1 (Perimeter of ABC). 2 A B
P C
Q
R
3. The incircle of ABC touches the sides BC, CA and AB at D, E and F respectively. Show that AF + BD + CE = AE + BF + CD =
1 (Perimeter of ABC) 2
4. ABCD is a quadrilateral such that A = 90°. A circle with center O, touches the sides AB, BC, CD and DA at E, F, G, H respectively. If CD = 30 cm, AD = 15 cm and CF = 20 cm, find the radius of the circle. 5. A circle is inscribed in a ABC having sides 8 cm, 10 cm and 12 cm as shown. Find the length of AD, BE and CF.
c 12
E
F
D 10 cm
A
m
8c m
C
B
6. ABC is a right angle triangle, right angled at A, AB = 6 cm and AC = 8 cm. A circle with centre O is inscribed inside the triangle. Calculate the value of ‘r’, the radius of inscribed circle. 7. Circles are drawn from the three vertices of a ABC (as shown); taken as centre to touch each other externally. If the sides of the triangle are 4 cm, 6 cm and 8 cm, find the radii of the circles. A
C
B
8. In two concentric circles, prove that a chord of larger circle which is tangent to smaller is bisected at the point of contact. 9. Prove that tangents drawn at the ends of a chord of a circle make equal angles with chord. downloaded from www. amitbajajmaths.blogspot.com 1
10. P and Q are centres of circles of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm which touches the above circles externally. Given that PRQ = 90°, write an equation in x and solve it. 11. Two tangent segment BC, BD are drawn to a circle with centre O such that DBC = 120°. Prove that BO = 2.BC. 12. In the given figure, XP and XQ are two tangents to a circle with O from a point X outside the circle. ARB is tangent to circle at R. Prove that XA + AR = XB + BR X
R
A
B
P
Q O
13.The four sides AB, BC, CD and DA of a quadrilateral ABCD touches a circle at the points P, Q, R and S respectively. If AB = 6 cm, BC = 7 cm and CD = 4 cm, find AD. C
m 4c R
D 7 cm
Q S
A
B
P 6 cm
14. In the given figure, two circles with centres X and Y touch externally at P. If tangents AT and BT meet the common tangent at T, then prove that AT = BT. T
A B X
P
Y
15. The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of a bigger circle. BD is a tangent to the smaller circle, touching it at D. Find the length AD. A O
B
D
ANSWERS 1. 24 cm
4. 3 cm
7. 3 cm, 5 cm and 1 cm
5. AD = 3 cm, BE = 7 cm and CF= 5 cm 10. x2 + 11x – 102 = 0; x = 6
downloaded from www. amitbajajmaths.blogspot.com 2
6. 2 cm
13. 3 cm
15. 19 cm
CIRCLES
CLASS X
1. A point P is 26 cm away from the centre O of a circle and the length PQ of the tangent segment drawn from P to the circle is 10 cm. Find the radius of the circle. 2. A circle is touching the side BC of ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ =
1 (Perimeter of ABC). 2 A B
P C
Q
R
3. The incircle of ABC touches the sides BC, CA and AB at D, E and F respectively. Show that AF + BD + CE = AE + BF + CD =
1 (Perimeter of ABC) 2
4. ABCD is a quadrilateral such that A = 90°. A circle with center O, touches the sides AB, BC, CD and DA at E, F, G, H respectively. If CD = 30 cm, AD = 15 cm and CF = 20 cm, find the radius of the circle. 5. A circle is inscribed in a ABC having sides 8 cm, 10 cm and 12 cm as shown. Find the length of AD, BE and CF.
c 12
E
F
D 10 cm
A
m
8c m
C
B
6. ABC is a right angle triangle, right angled at A, AB = 6 cm and AC = 8 cm. A circle with centre O is inscribed inside the triangle. Calculate the value of ‘r’, the radius of inscribed circle. 7. Circles are drawn from the three vertices of a ABC (as shown); taken as centre to touch each other externally. If the sides of the triangle are 4 cm, 6 cm and 8 cm, find the radii of the circles. A
C
B
8. In two concentric circles, prove that a chord of larger circle which is tangent to smaller is bisected at the point of contact. 9. Prove that tangents drawn at the ends of a chord of a circle make equal angles with chord. downloaded from www. amitbajajmaths.blogspot.com 1
10. P and Q are centres of circles of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm which touches the above circles externally. Given that PRQ = 90°, write an equation in x and solve it. 11. Two tangent segment BC, BD are drawn to a circle with centre O such that DBC = 120°. Prove that BO = 2.BC. 12. In the given figure, XP and XQ are two tangents to a circle with O from a point X outside the circle. ARB is tangent to circle at R. Prove that XA + AR = XB + BR X
R
A
B
P
Q O
13.The four sides AB, BC, CD and DA of a quadrilateral ABCD touches a circle at the points P, Q, R and S respectively. If AB = 6 cm, BC = 7 cm and CD = 4 cm, find AD. C
m 4c R
D 7 cm
Q S
A
B
P 6 cm
14. In the given figure, two circles with centres X and Y touch externally at P. If tangents AT and BT meet the common tangent at T, then prove that AT = BT. T
A B X
P
Y
15. The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of a bigger circle. BD is a tangent to the smaller circle, touching it at D. Find the length AD. A O
B
D
ANSWERS 1. 24 cm
4. 3 cm
7. 3 cm, 5 cm and 1 cm
5. AD = 3 cm, BE = 7 cm and CF= 5 cm 10. x2 + 11x – 102 = 0; x = 6
downloaded from www. amitbajajmaths.blogspot.com 2
6. 2 cm
13. 3 cm
15. 19 cm
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