Question 12 of 50intermediate🔧 ApplyDiagram4 marks

Draw ΔABC\Delta ABC, with AB=6AB = 6 cm, BC=7BC = 7 cm and CA=7CA = 7 cm. Draw the circumcircle of ΔABC\Delta ABC. Let the circumcentre be O. Measure OA, OB, OC.

Correct Answer

OA = OB = OC ≈\approx 3.9 cm

Exercise: Exercise Set 5.1 | Q: 3 | (Page 7)
For More Understanding

Explanation

To draw the circumcircle of a triangle, we must find its circumcentre, which is the intersection point of the perpendicular bisectors of the triangle's sides. The context explicitly states that the intersection of the perpendicular bisectors of AB and BC is a single point, meaning there is exactly one circumcircle for a given triangle. Because the given triangle with sides 6 cm, 7 cm, and 7 cm is an acute-angled triangle, the circumcentre will lie inside the triangle. By definition, the circumcentre is equidistant from all vertices, making OA, OB, and OC the radii of the circumcircle, hence they are equal in length.

Solution Steps

  1. Step 1: Draw ΔABC\Delta ABC with the given dimensions: AB=6AB = 6 cm, BC=7BC = 7 cm, and CA=7CA = 7 cm. The triangle is acute-angled and isosceles.

  2. Step 2: Construct the perpendicular bisectors of at least two sides of the triangle, such as AB and BC.

  3. Step 3: Mark the point of intersection of these perpendicular bisectors as O. This point O is the circumcentre. Since the triangle is acute-angled, O lies inside the triangle.

  4. Step 4: With O as the centre and radius equal to the distance from O to any vertex (say OA), draw the circumcircle passing through A, B, and C.

  5. Step 5: Measure the lengths OA, OB, and OC. Since they are all radii of the same circumcircle, OA = OB = OC, and each measures approximately 3.9 cm.

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