Question : $\triangle ABC$ is an isosceles triangle with AB = AC = 15 cm and an altitude from A to BC of 12 cm. The length of side BC is:

Option 1: 9 cm

Option 2: 12 cm

Option 3: 18 cm

Option 4: 20 cm

Question : ABC is an isosceles right-angle triangle. $\angle ABC = 90 ^{\circ}$ and AB = 12 cm. What is the ratio of the radius of the circle inscribed in it to the radius of the circle circumscribing $\triangle ABC$?

Option 1: $6–\sqrt{2}: 3 \sqrt{2}$

Option 2: $2–\sqrt{2}: \sqrt{2}$

Option 3: $6–3 \sqrt{2}: 1 \sqrt{2}$

Option 4: $6–3 \sqrt{2}: 6 \sqrt{2}$

Question : $O$ is the circumcentre of the isosceles $\triangle ABC$. Given that $AB = AC = 5$ cm and $BC = 6$ cm. The radius of the circle is:

Option 1: 3.015 cm

Option 2: 3.205 cm

Option 3: 3.025 cm

Option 4: 3.125 cm

Question : A circle is inscribed in a ΔABC having sides AB = 16 cm, BC = 20 cm, and AC = 24 cm, and sides AB, BC, and AC touch circle at D, E, and F, respectively. The measure of AD is:

Option 1: 10 cm

Option 2: 20 cm

Option 3: 6 cm

Option 4: 14 cm

Question : In a triangle ${ABC}, {AB}={AC}$ and the perimeter of $\triangle {ABC}$ is $8(2+\sqrt{2}) $ cm. If the length of ${BC}$ is $\sqrt{2}$ times the length of ${AB}$, then find the area of $\triangle {ABC}$.

Option 1: $32 \ cm^2$

Option 2: $28 \ cm^2$

Option 3: $16 \ cm^2$

Option 4: $36 \ cm^2$