Question : If the square of the product of three sides of the triangle is 1024 units and the area of its circumcircle is $16 \pi$, what is the area of the given triangle?
Option 1: 2.5 sq. units
Option 2: 3 sq. units
Option 3: 2 sq. units
Option 4: 4 sq. units
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Correct Answer: 2 sq. units
Solution : Area of circle = $\pi R^2$ where $R$ is the radius of the circumcircle. ⇒ $16\pi = \pi R^2$ ⇒ $R = 4$ The length of the three sides is $a,b$, and $c$. The area of the triangle is denoted by $A$. $(abc)^2 = 1024$ ⇒ $(abc) = 32$ Using $R = \frac{abc}{4A}$ ⇒ $4 = \frac{32}{4A}$ ⇒ $A = 2$ Hence, the correct answer is 2 sq. units.
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Question : If the numerical value of the circumference and area of a circle is the same, then the area is:
Option 1: $6\pi$ sq. units
Option 2: $4\pi$ sq. units
Option 3: $8\pi$ sq. units
Option 4: $12\pi$ sq. units
Question : $\triangle$ABC is similar to $\triangle$PQR. The length of AB is 16 cm and the length of the corresponding side PQ is 9 cm. If the area of $\triangle$ABC is 1024 sq. cm, what is the area of $\triangle$PQR?
Option 1: 768 sq. cm
Option 2: 32 sq. cm
Option 3: 324 sq. cm
Option 4: 128 sq. cm
Question : Two triangles MNO and XYZ are given similar with the ratio of their side as 9 : 4. If the area of the larger triangle is 243 sq. cm, then the area of the smaller triangle will be:
Option 1: 162 sq. cm
Option 2: 28 sq. cm
Option 3: 48 sq. cm
Option 4: 16 sq. cm
Question : The length of a side of a square inscribed in a circle is $a\sqrt2$ units. The circumference of the circle is:
Option 1: $2\pi a$ units
Option 2: $\pi a$ units
Option 3: $4\pi a$ units
Option 4: $\frac{2a}{\pi}$ units
Question : The three sides of a triangle are 7 cm, 9 cm, and 8 cm. What is the area of the triangle?
Option 1: $12 \sqrt{3} \;\mathrm{Sq} . \mathrm{cm}$
Option 2: $10\sqrt{3} \;\mathrm{Sq} . \mathrm{cm}$
Option 3: $12 \sqrt{5} \;\mathrm{Sq} . \mathrm{cm}$
Option 4: $2 \sqrt{5} \;\mathrm{Sq} . \mathrm{cm}$
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