Question : In the given figure, the radius of a circle is $14\sqrt{2}$ cm. PQRS is a square. EFGH, ABCD, WXYZ, and LMNO are four identical squares. What is the total area (in cm2) of all the small squares?
Option 1: 31.36
Option 2: 125.44
Option 3: 62.72
Option 4: 156.8
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Correct Answer: 125.44
Solution :
Given that the radius of the circle = $14\sqrt{2}$ cm The diameter of the circle = $2 \times 14\sqrt{2} = 28\sqrt{2}$ cm. The side of the larger square (PQRS) = $\frac{\operatorname{diameter}}{\sqrt2}=28$ cm. TO = half the side of the larger square = $14$ cm. The radius (OX = OU) of the circle = $14\sqrt{2}$ cm Let the V as the midpoint of the side WX of one of the smaller squares. Let the side of this smaller square = $a$ cm ⇒ XY = WX = VT = $a$ cm ⇒ VX = $\frac{a}{2}$ cm ⇒ VO = VT + TO = $14 + a$. Using the Pythagorean theorem in triangle OVX, ⇒ OX2 = VX2 + VO2 ⇒ $(14\sqrt{2})^2 = (\frac{a}{2})^2 + (14 + a)^2$ ⇒ $ 392 = \frac{a^2}{4} + a^2 + 28a + 196 $ Multiply the entire equation by 4 $⇒ 1568 = a^2 + 4a^2 + 112a + 784 $ $⇒ 5a^2+112a-784=0$ $⇒ 5a^2+140a-28a-784=0$ $⇒ 5a(a+28)-28(a+28)=0$ $⇒ (a+28)(5a-28)=0$ $⇒ (5a-28)=0$ $⇒ a = 5.6$ cm The total area of all the small squares = $4 \times (5.6)^2 = 125.44$ cm2 Hence, the correct answer is 125.44.
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Question : In the given figure, ABCD is a square. EFGH is a square formed by joining mid-points of sides of ABCD. LMNO is a square formed by joining mid-points of sides of EFGH. A circle is inscribed inside LMNO. If the area of a circle is 38.5 cm2 then what is the area (in cm2) of square ABCD?
Option 1: 98
Option 2: 196
Option 3: 122.5
Option 4: 171.5
Question : In the given figure, PQR is a quadrant whose radius is 7 cm. A circle is inscribed in the quadrant as shown in the figure. What is the area (in cm2) of the circle?
Option 1: $385-221\sqrt{2}$
Option 2: $308-154\sqrt{2}$
Option 3: $154-77\sqrt{2}$
Option 4: $462-308\sqrt{2}$
Question : In the given figure, $PQRS$ is a quadrilateral. If $QR = 18$ cm and $PS = 9$ cm, then, what is the area (in cm2) of quadrilateral $PQRS$?
Option 1: $\frac{(64\sqrt{3})}{3}$
Option 2: $\frac{(177\sqrt{3})}{2}$
Option 3: $\frac{(135\sqrt{3})}{2}$
Option 4: $\frac{(98\sqrt{3})}{3}$
Question : What is the perimeter of a square inscribed in a circle of radius 5 cm?
Option 1: $20 \sqrt{2}\ \mathrm{~cm}$
Option 2: $40\sqrt{2}\ \mathrm{~cm}$
Option 3: $30\sqrt{2}\ \mathrm{~cm}$
Option 4: $10\sqrt{2}\ \mathrm{~cm}$
Question : There is a wooden sphere of radius $6 \sqrt{3}$ cm. The surface area of the largest possible cube cut out from the sphere will be:
Option 1: $864$ cm2
Option 2: $464 \sqrt{3}$ cm2
Option 3: $462$ cm2
Option 4: $646\sqrt{3}$ cm2
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