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Construct the incircle and circumcircle of an equilateral ΔDSP 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.

Answer
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Hint: Incentre is the meeting point of angle bisectors of triangle while circumcenter is meeting point of perpendicular bisector of sides and in equilateral triangle all the centers lie on the same point.

Complete step-by-step answer:
Here we have to construct the incircle and circumcircle of an equilateral ΔDSP with side 7.5 cm. Also we have to find the ratio of circumradius to inradius.

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First of all we have to construct the equilateral ΔDSP with side 7.5 cm for which given steps should be followed.
1. Draw the line SP=7.5 cm.
2. Now, with S as center and 7.5 cm as radius, draw an arc above the line SP.
3. With P as a center and 7.5 cm as radius, cut an arc on the previous drawn arc and name the point D.
4. Join DS and DP. Therefore our equilateral ΔDSP is obtained.
Now to draw the circumcenter of ΔDSP, we need to follow the following steps.
1. As we know the circumcenter is a point of intersection of perpendicular bisectors of sides of a triangle. First, construct the perpendicular bisectors of sides.
2. We first have to mark the mid points of sides DP and SP which is at the length 7.52=3.75 cm that is A and B respectively.
3. Now with D as a center and radius greater than DA, draw two arcs on both sides of DP.
4. With the same radius and center as P, draw two arcs on both sides of DP cutting the previous arcs and name them X and Y. Join the X and Y. Therefore XY is a perpendicular bisector of DP.
5. Now with S as center and radius greater than SB, draw two arcs on both sides of SP.
6. With the same radius and center as P, draw two arcs on both sides of SP cutting the previous arcs and name them M and N. Join M and N. Therefore MN is a perpendicular bisector of SP.
7. The point of intersection of MN and XY is O, that is the circumcenter.
8. Circumradius of the given circle is OP=OD=OS=R.
9. Therefore, with Oas center and OPas radius, draw a circumcircle of ΔDSP.
10. Now measure the circumradius that is the length of OP, which comes out to be OP=R=4.33 cm.
Now to draw the incircle of ΔDSP, we need to follow the following steps.
1. Incircle is the point of intersection of angle bisectors of triangles.
2. As we know that, in an equilateral triangle, all the centers coincide. Therefore, with O as center and OA as radius, draw the incircle of ΔDSP as shown.
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3. Inradius of the given incircle is OA=OB=OC=r.
4. Now measure the inradius that is the length of OA, which comes out to be OA=r=2.165.
Therefore ratio radius of circumcircle to the radius of incircle =length of circumradiuslength of inradius=OPOA
Therefore we get, Rr=4.332.165=21=2:1
Note: Students can also first construct the angle bisectors and get the center with their intersection which is the same as O. Also, students must remember that incenter and circumcenter coincide only in equilateral triangles.