1 Million Means: 1 million in numerical is represented as 10,00,000. The Indian equivalent of a million is ten lakh rupees. It is not a...

1 Million Means: 1 Million in Rupees, Lakhs and Crores

June 5, 2024**Circumcircle of a Triangle: **The circle passing through all three vertices of a triangle is called the circumcircle of a triangle. The centre of that circle which passes through all the vertices of a triangle is called the circumcentre of the triangle. When the vertices of a triangle lie on a circle, the sides of the triangle form chords of the circle.

The radius of a circumcircle is equal to the distance between the circumcentre and any one of the triangle’s three vertices. Thus, the circumcentre of a triangle is equidistant from the three vertices. The centre is where three unique lines cross: lines at right angles to the midpoint of each side of the triangle.

The circumscribed circle or circumcircle of a triangle is a circle that passes through all the vertices of a triangle. The centre of this circle is called the circumcentre, and its radius is called the circumradius.

Example:

To construct a circle circumscribing a given triangle (say \(\Delta ABC\)), draw the perpendicular bisectors of any two sides of the triangle. Let these perpendicular bisectors meet at point \(O\). Taking point \(O\) as centre and \(OA\) or \(OB\) or \(OC\) as radius, draw a circle that will pass through all the triangle’s three vertices. Here, point \(O\) is called the circumcentre of the triangle, and \(OA = OB = OC\) is called its circumradius.

**Perpendicular Bisectors: **A perpendicular bisector is a line that forms a right angle with the given line segment and bisects the line segment into two equal parts. The perpendicular bisectors of the sides of a triangle are concurrent, i.e., they meet at one point.

The construction of the circumcircle of the triangle \(ABC\) is shown in the below steps.

Draw the perpendicular bisector of the side \(AB\)

Draw the perpendicular bisector of the side \(BC\)

Mark the point where the perpendicular bisectors intersect as \(O\). The triangles circumcentre.

With the compass on \(O\), set its width to \(C\) and draw a circle.

The circle above is the circumcircle of a triangle \(ABC\), which is touching all the vertices of the triangle \(ABC\).

In a right-angled triangle, the hypotenuse is a diameter of the circumcircle, and its centre is precisely at the midpoint of the hypotenuse. This is the same situation as Thales theorem, where the diameter subtends a right angle to any point on a circle’s circumference.

In the case of an acute-angled triangle, the circumcentre lies inside the triangle, as shown below.

In the case of an acute-angled triangle, the circumcentre lies outside the triangle, as shown below.

** Q.1. Using ruler and compass only, draw an equilateral triangle of side **\(4.5\,{\text{cm}}\)

Draw a line segment \(BC = 4.5\,{\text{cm}}\)

1. With centres \(B\) and \(C\), draw arcs of radius \(4.5\,{\text{cm}}\), each intersecting each other at \(A\)

2. Join \(AB\) and \(AC\). \(\Delta ABC\) is an equilateral triangle

3. Draw the bisectors of side \(BC\) and \(AC\) intersecting each other at \(O\)

4. Join \(OB\)

5. With centre \(O\) and radius \(OB\), draw a circle that passes through \(A, B\) and \(C\)

6. This is the required circle, and on measuring, the length of the radius is \(2.6\,{\text{cm}}\)

** Q.2. Construct a triangle with sides **\(5\,{\text{cm}}\), \(4\,{\text{cm}}\)

1. Draw a line segment \(BC = 5\,{\text{cm}}\)

2. With centre \(B\) and radius \(4\,{\text{cm}}\) and centre \(C\) and radius \(3\,{\text{cm}}\), draw arcs intersecting each other at \(A\)

3. Join \(AB\) and \(AC\)

4. Draw the perpendicular bisectors of sides \(AC\) and \(BC\) intersecting each other at \(O\)

5. We see \(O\) lies on the side \(BC\)

6. With centre \(O\) and radius \(OB\), draw a circle that will pass through \(A, B\) and \(C\), respectively. Which is the required circumcircle, and on measuring, the length of the radius is \(2.5\,{\text{cm}}\)

** Q.3. Using a ruler and a pair of compasses only, construct a triangle **\(ABC\)

Steps of construction:

1. Draw a line segment \(AB = 4\,{\text{cm}}\)

2. At \(B\), draw a ray \(BX\) making an angle of \(90^\circ \) and cut off \(BC = 6\,{\text{cm}}\)

3. Join \(AC\)

4. Draw the perpendicular bisectors of sides \(AB\) and \(AC\), which intersects each other at \(O\)

5. With centre \(O\), and radius equal to \(OB\) or \(OA\) or \(OC\) draw a circle which passes through \(A,B\) and \(C\)

This is the required circumcircle.

** Q.4. Draw a right-angled **\(\Delta ABC\)

1. Draw a line segment \(BC = 6.4\,{\text{cm}}\)

2. Draw its perpendicular bisector \(MX\) and cut off \(ML = 2.5\,{\text{cm}}\)

3. With the centre \(M\) and diameter \(BC\), draw a semi-circle.

4. Through \(L\), draw a parallel to \(BC\) intersecting the semi-circle at \(A\) and \(A’\)

5. Join \(AB\) and \(AC\)

6. Therefore, \(ABC\) is a right-angled triangle, right angles at \(A\).

7. \(M\) is the circumcentre

8. With centre \(M\) and radius \(BM\), complete the circle which passes through \(A, B\) and \(C\), which is the required circumcircle.

On measuring its radius is \(BM = \frac{1}{2}BC\)

\( = \frac{1}{2} \times 6.4~{\text{cm}} = 3.2~{\text{cm}}\)

** Q.5. Using the ruler and compass only, construct a **\(\Delta PQR\)

1. Draw a line segment \(QR = 6\,{\text{cm}}\)

2. At \(Q\) draw a ray making an angle of \(60^\circ \) and at \(R\), a ray making an angle of \(75^\circ \) which intersects each other at \(P\).

3. Draw the perpendicular bisector of side \(QR\) and \(PR\) intersecting each other at \(O\).

4. With centre \(O\) and radius \(OQ\) draw a circle. It will pass through \(P, Q\) and \(R\).

** Q.6. The sides **\(BC, CA, AB\)

1. The triangle \(ABC\) is constructed such that \(BC = 9\,{\text{cm}},\,CA = 6.5\,{\text{cm}}\), and \(AB = 8.5\,{\text{cm}}\).

2. Perpendicular bisectors of the sides \(AB\) and \(BC\) are drawn. They intersect at \(O\).

3. With \(O\) as centre and \(OA\) as radius, a circle is drawn. This is the required circumcircle.

4. By construction, \(O\) is equidistant from \(A, B\) and \(C\). So, the circle is drawn with \(O\) as the centre, and \(OA\) as radius will pass through all the three vertices \(A, B, C\) of the triangle \(ABC\).

From this article, we learned about the definition of the circumcircle of a triangle. Also, we learned the meaning of circumcentre, circumradius of a triangle. Also, we have known the position of circumcentre in different types of triangles and solved some construction problems based on the circumcircle of a triangle.

*Q.1. How do you find the circumcircle of a triangle?*** Ans:** The circle which passes through the vertices of a given triangle is known as the circumcircle of a triangle. So, if we want to fins the circumcircle, we want to verify whether the circle is passing through the vertices of a given triangle.

*Q.2. What is the formula of circumradius?** Ans:* The circumradius is given by \(R = \frac{{abc}}{{4rs}}\), where \(R\) is the circumradius, \(r\) is the inradius, \(a, b\) and \(c\) are the respective sides of the triangle.

*Q.3. How do you find the circumcentre of a triangle when the coordinates of the vertices of the triangle are given?** Ans:* If \(A\left({{x_1},{y_1}} \right),B\left({{x_2},{y_2}} \right)\) and \(C\left({{x_3},{y_3}} \right)\) are the vertices of a triangle \(ABC\). The steps to calculate the circumcentre of a triangle is:

1. Calculate the midpoint of given coordinates, i.e., midpoints of \(AB, AC\) and \(BC\)

2. Calculate the slope of the line

3. By using the midpoint and the slope, find out the equation of the line \(\left({y – {y_1}} \right) = m\left({x – {x_1}} \right)\)

4. Find out the equation of the other line in the same way

5. Solve two bisector equations by finding out the intersection point

6. The calculated intersection point will be the circumcentre of the given triangle

*Q.4. Where is the circumcentre of an obtuse-angled triangle located? ** Ans:* The circumcentre of an obtuse-angled triangle is located outside the triangle.

*Q.5. What is the circumcentre of a triangle?***Answer**: The circumcentre of a triangle is a point that is equidistant from the three vertices of the triangle.

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