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⭕ Chapter 5: I'm Up and Down, and Round and Round • Exercise Set 5.1

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Exercise Set 5.1 3Think, Draw and Infer 1Exercise Set 5.2 2Exercise Set 5.3 3Exercise Set 5.4 2Exercise Set 5.5 3Exercise Set 5.6 1End-of-Chapter Exercises 12All Exercises 27
Q1

Draw $\Delta ABC$ with $AB = 5\text{ cm}$, $\angle A = 70^\circ$ and $\angle B = 60^\circ$. Draw the circumcircle of $\Delta ABC$. Is the centre inside or outside the triangle?

Answer: Since $\angle A = 70^\circ$ and $\angle B = 60^\circ$, the third angle is $\angle C = 180^\circ - (70^\circ + 60^\circ) = 50^\circ$. Because all three angles ($70^\circ, 60^\circ, 50^\circ$) are less than $90^\circ$, $\Delta ABC$ is an acute-angled triangle. For an acute-angled triangle, the circumcentre lies inside the triangle.

O (Inside) A (70°) B (60°) C (50°) AB = 5 cm
Acute-angled ΔABC: Circumcentre O lies strictly inside the triangle
Q2

Draw $\Delta ABC$ with $AB = 5\text{ cm}$, $\angle A = 100^\circ$, $AC = 4\text{ cm}$. Draw the circumcircle of $\Delta ABC$. Is the centre inside or outside the triangle?

Answer: Since $\angle A = 100^\circ$, which is greater than $90^\circ$, $\Delta ABC$ is an obtuse-angled triangle. For an obtuse-angled triangle, the circumcentre lies outside the triangle.

O (Outside) A (100°) B C 5 cm 4 cm
Obtuse-angled ΔABC (∠A = 100°): Circumcentre O lies outside the triangle
Q4

What is the least possible radius of a circle through two points A and B?

Answer: The least possible radius of a circle passing through two points $A$ and $B$ occurs when the line segment $AB$ itself is the diameter of the circle. Therefore, the least possible radius is half the length of the segment $AB$ ($AB/2$).

A B O (Midpoint) r = AB/2
Minimum radius circle has segment AB as its diameter (r = AB / 2)