Question

Draw a circle of radius 3 centimetres and draw a rhombus with one angle 40°, all four sides touching the circle.

Answer 1

The rough sketch of the required figure can be drawn as follows:

We know that the central angle of the smaller arc between the two points on a circle and the angle between the tangents at these points are supplementary.

∠ECF + ∠EOF = 180°

⇒ ∠EOF = 180° − 40° = 140°

We also know that any tangent to circle is perpendicular to the radius to the point of contact.

∴ ∠OFC = ∠OGB = ∠OHD = ∠OEC = 90°

We will use these measurements to construct the required figure.

The steps of construction are as follows:

1) Draw a circle with centre O and radius, OF = 3 cm.

2) Draw ∠OFX of measure 90° at point F by taking OF as the base. Extend ray XF downwards.

3) Draw ∠EOF of measure 140° at point O by taking FO as the base.

4) Draw ∠OEC of measure 90° at point E by taking OE as the base. Extend CE downwards.

5) Extend the line segments EO and OF such that they intersect the circle at points G and H respectively.

6) Draw ∠OGB of measure 90° at point G by taking OG as the base, where B is a point on line FX. Extend BG downwards.

7) Draw ∠OHD of measure 90° at point H by taking OH as the base, where D is a point on ray CE.

8) Extend line segment DH to intersect ray BG at point A.

ABCD is the required rhombus with all the four sides touching the circle.