Fill in the blanks so as to make each of the following statement true : (Que. No. 7 to 12) 7. If the midpoint of \( \triangle \mathrm{ABC} \) is \( (3,-1) \) and \( \mathrm{A}(1,3) \) then the midpoint of \( \overline{\mathrm{BC}} \) is \( \qquad \) \( ((-3,4),(4,-3), \overline{(-4,3)}) \) 8. If \( \sin \theta=\sqrt{3} \cos \theta \) then \( \sec \theta= \qquad \) \( (2,3,1) \) 9. Quadrilateral ABCD circumscribes a circle. If \( \mathrm{AB}=5.2 \mathrm{~cm}, \mathrm{BC}=8.7 \mathrm{~cm} \) and \( C D=10.3 \mathrm{~cm} \), then \( \mathrm{AD}= \qquad \) cm . 10. The area of a square which is inside a circle of radius 8 cm is \( \qquad \) cm2. \( (64,60,128) \) 11. The volume of cylinder is 550 cubic cm . If the radius of it is 5 cm , then find the height \( = \qquad \) \( (7,6,5) \) 12. If the observations \( 24,25,26, x+2, x+3,30,31 \) and 34 are in ascending order, and the median is 27.5 then \( x= \qquad \) \( (27,25,28) \)

Let's solve each question step-by-step: --- **7. Midpoint of \(\overline{\mathrm{BC}}\)** Given: - Centroid (midpoint of the triangle) = \((3, -1)\) - Vertex \( \mathrm{A}(1, 3) \) The centroid of a triangle is the average of its vertices: \[ (3, -1) = \left( \frac{1 + B_x + C_x}{3}, \frac{3 + B_y + C_y}{3} \right) \] Solving for \( B_x + C_x \) and \( B_y + C_y \): \[ 1 + B_x + C_x = 9 \implies B_x + C_x = 8 \] \[ 3 + B_y + C_y = -3 \implies B_y + C_y = -6 \] The midpoint of \(\overline{\mathrm{BC}}\) is: \[ \left( \frac{B_x + C_x}{2}, \frac{B_y + C_y}{2} \right) = \left( \frac{8}{2}, \frac{-6}{2} \right) = (4, -3) \] **Answer:** \( (4, -3) \) --- **8. Value of \(\sec \theta\)** Given: \[ \sin \theta = \sqrt{3} \cos \theta \] Divide both sides by \(\cos \theta\): \[ \tan \theta = \sqrt{3} \implies \theta = 60^\circ \] \[ \sec \theta = \frac{1}{\cos \theta} = 2 \] **Answer:** \( 2 \) --- **9. Length of \(\mathrm{AD}\)** For a quadrilateral that circumscribes a circle (tangential quadrilateral), the sum of opposite sides are equal: \[ \mathrm{AB} + \mathrm{CD} = \mathrm{BC} + \mathrm{AD} \] Given: \[ 5.2 + 10.3 = 8.7 + \mathrm{AD} \implies 15.5 = 8.7 + \mathrm{AD} \implies \mathrm{AD} = 6.8 \text{ cm} \] **Answer:** \( 6.8 \) cm --- **10. Area of the Square Inside the Circle** Given: - Radius of the circle = 8 cm - Diagonal of the square = Diameter = \(16\) cm Using the diagonal to find the side \(s\) of the square: \[ s \sqrt{2} = 16 \implies s = \frac{16}{\sqrt{2}} = 8\sqrt{2} \text{ cm} \] Area: \[ s^2 = (8\sqrt{2})^2 = 128 \text{ cm}^2 \] **Answer:** \( 128 \) cm² --- **11. Height of the Cylinder** Given: - Volume \( V = 550 \) cm³ - Radius \( r = 5 \) cm Volume formula: \[ V = \pi r^2 h \implies 550 = \pi \times 25 \times h \implies h = \frac{550}{25\pi} = \frac{22}{\pi} \approx 7 \text{ cm} \] **Answer:** \( 7 \) --- **12. Value of \( x \)** Given observations in order: \[ 24, 25, 26, x+2, x+3, 30, 31, 34 \] Median is the average of the 4th and 5th numbers: \[ \frac{(x+2) + (x+3)}{2} = 27.5 \implies 2x + 5 = 55 \implies x = 25 \] **Answer:** \( 25 \) --- **Summary of Answers:** 7. \( (4, -3) \) 8. \( 2 \) 9. \( 6.8 \) cm 10. \( 128 \) cm² 11. \( 7 \) 12. \( 25 \)