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Constructions

Short Answer Type

151. Divide a line segment of 7 cm internally in the ratio 2 : 3

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152. Draw a circle of radius 3 cm. From a point 5 cm away from the centre of the circle, draw two tangents to the circle. Find the length of the tangents.

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153. Construct a ΔABC in which AB = 6.5 cm, ∠B = 60° and BC = 5.5 cm. Also, construct a triangle ABC similar to ΔABC, whose each sice

3 over 2

times the corresponding side of the ΔABC.

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154.

Construct a ΔABC in which AB = 6.5 cm, ∠B = 60° and BC = 5.5 cm. Also, construct a triangle ABC similar to ΔABC, whose each side $bold 3 over bold 2$ times the corresponding side of the ΔABC.

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155. Draw a ΔABC with sides BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Construct a ΔABC similar to ΔABC, such that sides of ΔABC are

bold 3 over bold 4

of the corresponding sides of ΔABC.

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156. Draw a circle of radius 4 cm. Form a point P, 7 cm from the centre of the circle, draw a pair of tangents to the circle. Measure the length of each tangent segment.

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157. Draw a right triangle in which the sides containing the right angles are 5 cm and 4 cm. Construct a similar triangle whose sides are

bold 5 over bold 3

times the sides of above.

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158.

Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60^o to each other.

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Long Answer Type

159.
Prove that the length of tangents drawn from an external point to a circle is equal.

Given: TP and TQ are two tangent drawn from an external point T to the circle C (O, r).

To prove: TP = TQ

Construction: Join OT

Proof: we know that a tangent to the circle is perpendicular to the radius through the point of contanct.

∴ ∠OPT = ∠OQT = 90^o

In Δ OPT and ΔOQT

OT = OT (common)

OP = OQ (radius of the circle)

∠OPT = ∠OQT (90^o)

∴ ∠OPT = ∠OQT (RHS congruence criterion)

⇒ TP = TQ (CPCT)

Hence, the length of the tangents drawn from an external point to a circle is equal.