Q1. To divide a line segment in 5:3, we
have to make
Solution :
To
divide a line segment in m:n,
we have to make m+n equal divisions.
Q2. Two tangents drawn from the
external points to a circle are
Solution :
Two tangents drawn from
the external points to a circle are equal.
Q3. To divide a line segment in m:n, we have to make
Solution :
To divide a line segment in m:n, we have m+n equal divisions.
Q4. The tangent is perpendicular to the
Solution :
The
tangent makes an angle of measure 90o with the radius and
diameter.
Q5. In the
figure below, ∠XAO = 30
°. What is the measure of ∠XOA?
Solution :
∠XAO = 30°
∠OXA = 90°
So, ∠XOA = 60° … (sum of the angles of a triangle)
Q6. If AB is
divided into 3 equal parts at P and R, then
Solution :
A-P-R-B
If PB =
2x, BA = 3x, AP = x, PR = RB = x.
Q7. The
length of the tangent from the external point P is 15 cm, and the distance of
P from the centre is 17 cm. So, the radius of the circle is
Solution :
By Pythagoras' theorem,
172 = 152 + radius2
radius2 = 64
radius = 8 cm
Q8. In the
figure below, if AO = 29 cm, XO = 21 cm, then AY =?
Solution :
Since the tangent is perpendicular to the
radius,
By Pythagoras' theorem,
292 = 212 + AX2
AX2 = 400
AY = AX = 20 cm
Q9. How many
tangents can be drawn to a circle having a diameter 15 cm from a point at a
distance of 7.5 cm from the centre?
Solution :
The distance from the centre is 7.5 cm.
The radius is 7.5 cm.
Since the point lies on the circle, 1
tangent can be drawn to the circle.
Q10. If the scale factor is less than
one, then the new triangle is
Construction of Similar Triangles" >
Solution :
If
the scale factor is less than one, then the new triangle is smaller than the original
one.
Q11. In the figure below, if ∠XAO = 30°, then what is
the measure of ∠XAY?
Solution :
The
line joining the centre of a circle to the external
point bisects the angle between two tangents and two radii.
2∠XAO = ∠XAY
Q12. The angle drawn in a semicircle from
the end points of the diameter is
Solution :
The
angle drawn in a semicircle from the end points of the diameter is 90°.
Q13. The
number of tangents which can be drawn through a point on the circle is
Solution :
Only one
tangent can be drawn through a point on the circle.
Q14. While
constructing a similar triangle such that its corresponding sides are in the
ratio 5:2, we need to construct an acute angle at any one of the bases of the
given triangle and make
Solution :
While
constructing a similar triangle, we need to construct an acute angle at any
one of bases of the given triangle and make 5 equal divisions (max (5 and
2)).
Q15. If PQ is
divided into 5 equal parts at A, B, C and D, then PB:BQ is
Solution :
P-A-B-C-D-Q,
PB = 2x, BQ = 3x
Q16. While
constructing ∆PQR similar to ∆ABC such that AB:PQ = 3:5, we need to make an acute
angle at A and divide it
Construction of Similar Triangles" >
Solution :
While
constructing a triangle similar to the given triangle, such that their sides
are in the ratio m:n, we need to make an acute angle
and divide it max (m,n) times.
Q17. A tangent cuts the circle at
Solution :
A
tangent cuts the circle at one point.
Q18. How many
tangents can be drawn to a circle having a diameter 12 cm from a point at a
distance of 10 cm from the centre?
Solution :
The distance from the centre is 10 cm.
The radius is 6 cm.
Since the point lies outside the circle, 2
tangents can be drawn to the circle.
Q19. The
number of tangents which can be drawn from a point inside a circle to it is
Solution :
No
tangents can be drawn from a point inside a circle.
Q20. In the figure below, AO = 5 cm and
XO = 3 cm, then AX =?
Solution :
As the radius is
perpendicular to the tangent,
By Pythagoras' theorem,
52 = 32
+ AX2AX2 = 16
AX = 4
