Construct incircle and circumcircle of an equilateral

Δ DSP with side 7.5 cm. Measure the radii of both the circles and find the ratio of radius of circumcircle to the radius of incircle.

#### Solution

**Steps of construction:**

1. Draw a line SP = 7.5 cm.

2. With S as centre and 7.5 cm as radius, draw an arc above the line SP.

3. With P as centre and 7.5 cm as radius, cut an arc on on the previous drawn arc and name the point of intersection as D.

4. Join DS and DP. Δ DSP is thus obtained.

5. Draw the perpendicular bisectors of the lines DP and SP. Let these perpendicular bisectors meet at point O.

4. With O as centre and OP as radius, construct a circle touching all the vertices of the Δ DSP.

This circle is thus the required circumcircle.

Radius = 4 cm

**Steps of construction**

1. Draw a line SP = 7.5 cm.

2. With S as centre and 7.5 cm as radius, draw an arc above the line SP.

3. With P as centre and 7.5 cm as radius, cut an arc on on the previous drawn arc and name the point of intersection as D.

4. Join DS and DP. Δ DSP is thus obtained.

5. Draw the angle bisectors of angle S and P and let them meet at point O.

6. Draw the perpendicular from point O to the line SP. Join OA.

7. With O as centre and OA as radius, draw a circle touching all the sides of the triangle.

This is the required incircle.

Radius = 2 cm

ratio of radius of circumcircle to the radius of incircle = 4 : 2 = 2 : 1