Skip to content. Change Language. Related Articles. Table of Contents. Save Article. Improve Article. Like Article. Last Updated : 15 Mar, Python 3 Program to find the area of the circle. This code is contributed by ChitraNayal. Pow b, 2. Pow b, 2 ;. WriteLine circlearea a, b ;. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Area of a circle inscribed in a rhombus? Ask Question. Asked 8 years, 1 month ago. Active 1 year, 6 months ago. Viewed 8k times. Phaptitude Phaptitude 2, 5 5 gold badges 18 18 silver badges 38 38 bronze badges. Add a comment. At the intersection of its diagonals. This is correct. Now, answer the question: why the center of the inscribed circle is located at the intersection of the rhombus diagonals?
I have an idea! Figure 1. To the Problem 1 If the circle is inscribed in the rhombus, it is inscribed in each of four of the rhombus interior angles. Therefore, the center of the inscribed circle is located at the angle bisectors of the rhombus interior angles. But the angle bisector of the rhombus is its diagonal. Hence, the center of the inscribed circle lies at the intersection point of the rhombus diagonals. You got the major idea to solve this construction problem. To complete this part of the lesson, I'd like to give the references to the lessons where the relevant geometric facts were proved.
The fact that the center of the circle inscribed in the angle is located at the angle bisector was proved in the lesson An angle bisector properties under the topic Triangles of the section Geometry in this site.
The fact that in the rhombus the diagonals are the angle bisectors was proved in the lesson Diagonals of a rhombus bisect its angles under the topic Parallelograms of the section Geometry in this site. Problem 2 Construct the radius of the circle inscribed in the given rhombus. Solution If the circle is inscribed in the rhombus, then the rhombus side is tangent to the circle.
It is visually clear from the Figure 2. Let us accept it without the formal proof now. It will be proved later in one of the lessons that follow. Therefore, in order to construct the radius of the inscribed circle, we should to construct the perpendicular from the center of the circle to the rhombus side. It doesn't matter to which one of four sides of the rhombus. To any. Figure 2. To the Problem 2 In other words, we should to construct the perpendicular from the rhombus center, which is the diagonal intersection point, to the rhombus side.
The algorithm of constructing the perpendicular from the point outside the straight line to the line using the ruler and the compass was described in the lesson How to bisect a segment using a compass and a ruler.
This lesson is under the topic Triangles of the section Geometry in this site. Thus you have the algorithm to construct the radius of the inscribed circle to the rhombus. The construction Problem 2 is solved.
As the last step in this lesson, I'd like you to solve the following Problem 3 Calculate the radius of the circle inscribed in the given rhombus.
The rhombus is given by its side measure and its diagonals measures and. Solution As we agreed above in the solution of the Problem 2 , the radius of the circle inscribed in the given rhombus is the perpendicular drawn from the center of the circle which is the diagonals intersection point to the side of the rhombus at the tangent point.
The diagonals of the rhombus are perpendicular and bisect each other. This was proved in the lessons Diagonals of a rhombus are perpendicular and Properties of diagonals of parallelograms under the topic Parallelograms of the section Geometry in this site.
Therefore, the diagonals of the rhombus divide it in four congruent Figure 3. To the Problem 3 right triangles, and we need to find the length of the altitude of any of four right triangles drawn from the right angle vertex to the hypotenuse.
So, our original problem is reduced to the following one: find the altitude of the right triangle drawn from the right angle vertex to the hypotenuse. The parameters of this right triangle are its legs measures a and b and the hypotenuse measure c Figure 4 linked to the parameters of the original rhombus by the equalities , ,.
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