Draw with , , . Draw the circumcircle of . Is the centre inside or outside the triangle?

Understand the Question To construct the triangle , we use the given side lengths , , and included angle . The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of its sides, and the circumcircle is drawn with the circumcentre as the centre passing through all three vertices. Since , is an obtuse-angled triangle, meaning its circumcentre lies outside the triangle. Step 1 · Draw the Base Draw a line segment . Step 2 · Construct Angle A Using a protractor at point , construct and draw ray . Step 3 · Locate Point C and Form Triangle ABC With as centre and radius , cut an arc on ray at point . Join to complete . Step 4 · Draw the Perpendicular Bisector of AB With centres and and a compass radius greater than , draw arcs on both sides of intersecting each other. Draw a line passing through these intersection points to obtain the perpendicular bisector of . Step 5 · Draw the Perpendicular Bisector of AC With centres and and a compass radius greater than , draw arcs on both sides of intersecting each other. Draw a line through these points to obtain the perpendicular bisector of . Step 6 · Locate the Circumcentre Let the perpendicular bisectors of and intersect at point . Point is the circumcentre of . Step 7 · Draw the Circumcircle With as centre and radius equal to (or or ), draw a circle. This circle passes through all three vertices , , and . Step 8 · Determine the Position of the Centre Since , is an obtuse-angled triangle. Therefore, the circumcentre lies outside the triangle. Answer > The centre is outside the triangle. Common Mistakes Position of Circumcentre: The circumcentre lies inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle. Since , the centre must be outside. Bisector Radius: Drawing arcs with a radius less than or equal to half the side length will prevent the arcs from intersecting. Incircle vs. Circumcircle: Confusing the circumcircle (intersection of perpendicular bisectors of sides) with the incircle (intersection of angle bisectors).

Circles and Geometric Shapes | Exercise 5.1

Question 2

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

Solution

Understand the Question

To construct the triangle $\Delta \text{ABC}$ , we use the given side lengths $\text{AB} = 5\text{ cm}$ , $\text{AC} = 4\text{ cm}$ , and included angle $\angle \text{A} = 100^\circ$ .
The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of its sides, and the circumcircle is drawn with the circumcentre as the centre passing through all three vertices.
Since $\angle \text{A} = 100^\circ > 90^\circ$ , $\Delta \text{ABC}$ is an obtuse-angled triangle, meaning its circumcentre lies outside the triangle.

Step 1 · Draw the Base

Draw a line segment $\text{AB} = 5\text{ cm}$ . Diagram 1

Step 2 · Construct Angle A

Using a protractor at point $\text{A}$ , construct $\angle \text{BAX} = 100^\circ$ and draw ray $\text{AX}$ . Diagram 2

Step 3 · Locate Point C and Form Triangle ABC

With $\text{A}$ as centre and radius $4\text{ cm}$ , cut an arc on ray $\text{AX}$ at point $\text{C}$ . Join $\text{BC}$ to complete $\Delta \text{ABC}$ . Diagram 3

Step 4 · Draw the Perpendicular Bisector of AB

With centres $\text{A}$ and $\text{B}$ and a compass radius greater than $\frac{1}{2}\text{AB}$ , draw arcs on both sides of $\text{AB}$ intersecting each other. Draw a line passing through these intersection points to obtain the perpendicular bisector of $\text{AB}$ . Diagram 4

Step 5 · Draw the Perpendicular Bisector of AC

With centres $\text{A}$ and $\text{C}$ and a compass radius greater than $\frac{1}{2}\text{AC}$ , draw arcs on both sides of $\text{AC}$ intersecting each other. Draw a line through these points to obtain the perpendicular bisector of $\text{AC}$ . Diagram 5

Step 6 · Locate the Circumcentre

Let the perpendicular bisectors of $\text{AB}$ and $\text{AC}$ intersect at point $\text{O}$ . Point $\text{O}$ is the circumcentre of $\Delta \text{ABC}$ . Diagram 6

Step 7 · Draw the Circumcircle

With $\text{O}$ as centre and radius equal to $\text{OA}$ (or $\text{OB}$ or $\text{OC}$ ), draw a circle. This circle passes through all three vertices $\text{A}$ , $\text{B}$ , and $\text{C}$ . Diagram 7

Step 8 · Determine the Position of the Centre

Since $\angle \text{A} = 100^\circ > 90^\circ$ , $\Delta \text{ABC}$ is an obtuse-angled triangle.

Therefore, the circumcentre $\text{O}$ lies outside the triangle.

Answer

The centre is outside the triangle.

Common Mistakes

Position of Circumcentre: The circumcentre lies inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle. Since $\angle \text{A} = 100^\circ$ , the centre must be outside.
Bisector Radius: Drawing arcs with a radius less than or equal to half the side length will prevent the arcs from intersecting.
Incircle vs. Circumcircle: Confusing the circumcircle (intersection of perpendicular bisectors of sides) with the incircle (intersection of angle bisectors).

Question 2

More questions in Exercise 5.1