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 90^{o} 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,
17^{2} = 15^{2} + radius^{2}
radius^{2} = 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,
29^{2} = 21^{2} + AX^{2}
AX^{2} = 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,
5^{2} = 3^{2}
+ AX^{2}AX^{2} = 16
AX = 4