Question 3

The circumcenter of a triangle is equidistant from its vertices.

Step 1 — Constructing the triangle

Let's start by drawing the base. Draw a line segment $\text{AB}$ of length 6 cm. Now, we will locate point C. With A as the center, draw an arc with a radius of 7 cm. With B as the center, draw another arc with a radius of 7 cm. These two arcs intersect at point C. Join $\text{AC}$ and $\text{BC}$. We have now formed $\Delta\text{ABC}$.

Step 2 — Finding the circumcentre

We need to find the circumcenter O. Draw the perpendicular bisector of side $\text{AB}$. Draw the perpendicular bisector of side $\text{BC}$. Draw the perpendicular bisector of side $\text{AC}$. These three bisectors meet at a single point. Let this intersection point be O. Point O is the circumcenter of $\Delta\text{ABC}$.

Step 3 — Drawing the circumcircle

Now, let's draw the circumcircle. With O as the center, take $\text{OA}$ as the radius. Draw a circle passing through A, B, and C. This circle is the circumcircle of $\Delta\text{ABC}$.

Step 4 — Measuring the radii

We measure the lengths $\text{OA}$, $\text{OB}$, and $\text{OC}$. We find that $\text{OA}$ is approximately 4 cm. $\text{OB}$ is approximately 4 cm. $\text{OC}$ is approximately 4 cm. All three lengths are equal, as expected for a circumcenter.

Answer

(i) $\text{OA} \approx \text{4 cm}$ (ii) $\text{OB} \approx \text{4 cm}$ (iii) $\text{OC} \approx \text{4 cm}$