Construct the incircle and circumcircle of an equilateral with side . 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 with side . Also we have to find the ratio of circumradius to inradius.
First of all we have to construct the equilateral with side for which given steps should be followed.
1. Draw the line .
2. Now, with as center and as radius, draw an arc above the line .
3. With as a center and as radius, cut an arc on the previous drawn arc and name the point .
4. Join and . Therefore our equilateral is obtained.
Now to draw the circumcenter of , 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 and which is at the length that is and respectively.
3. Now with as a center and radius greater than , draw two arcs on both sides of .
4. With the same radius and center as , draw two arcs on both sides of cutting the previous arcs and name them and . Join the and . Therefore is a perpendicular bisector of .
5. Now with as center and radius greater than , draw two arcs on both sides of .
6. With the same radius and center as , draw two arcs on both sides of cutting the previous arcs and name them and . Join and . Therefore is a perpendicular bisector of .
7. The point of intersection of and is , that is the circumcenter.
8. Circumradius of the given circle is .
9. Therefore, with as center and as radius, draw a circumcircle of .
10. Now measure the circumradius that is the length of , which comes out to be .
Now to draw the incircle of , 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 as center and as radius, draw the incircle of as shown.
3. Inradius of the given incircle is .
4. Now measure the inradius that is the length of , which comes out to be .
Therefore ratio radius of circumcircle to the radius of incircle
Therefore we get,
Note: Students can also first construct the angle bisectors and get the center with their intersection which is the same as . Also, students must remember that incenter and circumcenter coincide only in equilateral triangles.
Complete step-by-step answer:
Here we have to construct the incircle and circumcircle of an equilateral
First of all we have to construct the equilateral
1. Draw the line
2. Now, with
3. With
4. Join
Now to draw the circumcenter of
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
3. Now with
4. With the same radius and center as
5. Now with
6. With the same radius and center as
7. The point of intersection of
8. Circumradius of the given circle is
9. Therefore, with
10. Now measure the circumradius that is the length of
Now to draw the incircle of
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
3. Inradius of the given incircle is
4. Now measure the inradius that is the length of
Therefore ratio radius of circumcircle to the radius of incircle
Therefore we get,
Note: Students can also first construct the angle bisectors and get the center with their intersection which is the same as
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