Fill in the blanks so as to make each of the following statement true : (Que. No. 7 to 12) 7. If the midpoint of $$ riangle \mathrm{ABC} $$ 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 $$ \sin heta=\sqrt{3} \cos heta $$ then $$ \sec heta= \qquad $$ $$ (2,3,1) $$ 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 $$ x= \qquad $$ $$ (27,25,28) $$

Question

Fill in the blanks so as to make each of the following statement true : (Que. No. 7 to 12) 7. If the midpoint of $$ 	riangle \mathrm{ABC} $$ 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 $$ \sin 	heta=\sqrt{3} \cos 	heta $$ then $$ \sec 	heta= \qquad $$ $$ (2,3,1) $$ 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 $$ x= \qquad $$ $$ (27,25,28) $$

UpStudy AI · Accepted Answer

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. $$ 25 $$
7. $$ (4, -3) $$ 8. $$ 2 $$ 9. $$ 6.8 $$ cm 10. $$ 128 $$ cm² 11. $$ 7 $$ 12. $$ 25 $$

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Geometry Questions