Mathematics [3 marks sum]

BC = 4.2 cm, ∠ ABC = 60°, and AB = 5 cm.
Steps of construction:
(i) Draw BC of length 4.2 cm.
(ii) Draw an angle of 60° at B.
(iii) Cut BA = 5 cm and join A to B.
(iv) Draw angle bisector of ∠ ABC.
(v) Draw BD at 2 cm intersecting EF at O.
(vi) Taking O as centre and 2 cm as radius draw the required circle.

Steps of Construction:
(i) Draw a chord PQ through the given point on the circle.
(ii) Take a point R on the circle and join P and Q to a point R.
(iii) Construct ∠ QPY = ∠ PRQ on the opposite sides of the chord PQ.
(iv) Produce YP to X' to get YPX as the required tangent.
Construct a tangent to the circle from an external point:
In this section we shall study the construction at tangent to a circle from an external point when its center is;
(i) Know (ii) Unknown
Type (I). Construction at tangent to a circle from an external point when its centre is known.
Steps of Construction:
(i) Join the centre O of the circle to the given external point P i.e., join OP.
(ii) Draw right bisector of OP, intersecting OP at Q.
(iii) Taking Q as centre and OQ = PQ as radius, draw a circle to intersect the given circle at T and T'.

(iv) Join PT and PT' to get the required tangents as PT and PT'.

BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm

Radius of the circle = 2.5 cm

(i) See Figure,
(ii) Inside the triangle point, P is equidistant from BA and BC.
(iii) Line EF is the locus of points inside the triangle which are equidistant from B and C.
(iv) PB = 3.5 cm.

Steps of construction:

1) Draw a circle with centre O and radius = 2.5 cm.
2) With radius ( = 2.5 cm) cut off six equal arcs along the circumference and take these points as A, B, C, D, E, and F respectively.
3) Draw tangents at A, B, C, D, E, and F meeting to form circumscribed hexagon PQRSTU.

Steps of construction:

(i) Draw a circle with radius = 3.5 cm
(ii) Again make ∠ BOC = 120°
(iii) Draw diameter BOD and construct ∠ BOA = 90°
(iv) Join AB, AC, and BC. Then ∠ ABC is the required triangle.

Steps of Construction:
(i) Draw a line AX perpendicular to AB.
(ii) Mark off a point D on AX such that AD = 3.5 cm.
(iii) At D, draw the line DY at right angles to AX. Then DY is the required locus of the centre of circle that touches AB and has a radius of 3.5 cm.
(iv) Construct the bisector AZ of ∠ BAC intersecting DY at P.
(v) Draw PL, PM perpendicular to AB and AC respectively.
(vi) With P as centre and radius equal to 3.5, draw the circle which will pass through L and M.
Then this is the required circle that touches both AB and AC, and whose centre lies within the ∠ BAC.