The sides of a rectangle are in the ratio 2 : 3,
Question: The sides of a rectangle are in the ratio 2 : 3, and its perimeter is 20 cm. Draw the rectangle. Solution: Let the side be $x \mathrm{~cm}$ and $y \mathrm{~cm}$. So, we have : $2(x+y)=20$ Sides are in the ratio $2: 3$. $\therefore y=\frac{3 x}{2}$ Putting the value of $y$ : $2\left(x+\frac{3 x}{2}\right)=20$ $\frac{2 x+3 x}{2}=10$ $5 x=20$ $x=4$ $\therefore y=\frac{3 \times 4}{2}=6$ Thus, sides of the rectangle will be $4 \mathrm{~cm}$ and $6 \mathrm{~cm}$. $\mathrm{ABCD}$ is the recta...
Read More →In a rectangle ABCD,
Question: In a rectangleABCD, prove that ∆ACB ∆CAD. Solution: In ∆ACBand ∆CAD: AB = CD (rectangle property) AD = BC (rectangle property) AC ( common side ) Hence, by SSS criterion, it is proved that∆ACB ∆CAD....
Read More →A window frame has one diagonal longer than the other.
Question: A window frame has one diagonal longer than the other. Is the window frame a rectangle? Why or why not? Solution: No, since diagonals of a rectangle are equal....
Read More →Fill in the blanks in each of the following, so as to make the statement true:
Question: Fill in the blanks in each of the following, so as to make the statement true: (i) A rectangle is a parallelogram in which ..... (ii) A square is a rhombus in which ..... (iii) A square is a rectangle in which ..... Solution: (i) A rectangle is a parallelogram in whicheach angle is a right angle.(ii) A square is a rhombus in whicheach angle is a right angle.(iii) A square is a rectangle inwhich the adjacent sides are equal....
Read More →Which of the following statements are true for a square?
Question: Which of the following statements are true for a square? (i) It is a rectangle. (ii) It has all its sides of equal length. (iii) Its diagonals bisect each other at right angle. (iv) Its diagonals are equal to its sides. Solution: (i) True (ii) True (iii) True (iv) False This is because the hypotenuse in any right angle triangle is always greater than its two sides....
Read More →Which of the following statements are true for a rectangle?
Question: Which of the following statements are true for a rectangle? (i) It has two pairs of equal sides. (ii) It has all its sides of equal length. (iii) Its diagonals are equal. (iv) Its diagonals bisect each other. (v) Its diagonals are perpendicular. (vi) Its diagonals are perpendicular and bisect each other. (vii) Its diagonals are equal and bisect each other. (viii) Its diagonals are equal and perpendicular, and bisect each other. (ix) All rectangles are squares. (x) All rhombuses are par...
Read More →Show that each of the following systems of linear equations is consistent and also find their solutions:
Question: Show that each of the following systems of linear equations is consistent and also find their solutions: (i) $6 x+4 y=2$ $9 x+6 y=3$ (ii) $2 x+3 y=5$ $6 x+9 y=15$ (iii) $5 x+3 y+7 z=4$ $3 x+26 y+2 z=9$ $7 x+2 y+10 z=5$ (iv) $x-y+z=3$ $2 x+y-z=2$ $-x-2 y+2 z=1$ (v) $x+y+z=6$ $x+2 y+3 z=14$ $x+4 y+7 z=30$ (vi) $2 x+2 y-2 z=1$ $4 x+4 y-z=2$ $6 x+6 y+2 z=3$ Solution: (i) Here, $6 x+4 y=2 \quad \cdots(1)$ $9 x+6 y=3 \quad \cdots(2)$ $A X=B$ Here, $A=\left[\begin{array}{ll}6 4 \\ 9 6\end{arr...
Read More →Assertion (A) The curved surface area of a cone of base radius 3 cm and height 4 cm is 15π cm2.
Question: Assertion (A)The curved surface area of a cone of base radius 3 cm and height 4 cm is 15 cm2.Reason (R) Volume of a cone $=\pi r^{2} h$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).(c) Assertion (A) is true and Reason (R) is false.(d) Assertion (A) is false and Reason (R) is true. Solution: (c) Assertion (A) is true and R...
Read More →Assertion (A) The curved surface area of a cone of base radius 3 cm and height 4 cm is 15π cm2.
Question: Assertion (A)The curved surface area of a cone of base radius 3 cm and height 4 cm is 15 cm2.Reason (R) Volume of a cone $=\pi r^{2} h$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).(c) Assertion (A) is true and Reason (R) is false.(d) Assertion (A) is false and Reason (R) is true. Solution: (c) Assertion (A) is true and R...
Read More →Assertion (A) If the volumes of two spheres are in the ratio 27 : 8,
Question: Assertion (A)If the volumes of two spheres are in the ratio 27 : 8, then their surface areas are in the ratio 3 : 2.Reason (R) Volume of a sphere $=\frac{4}{3} \pi R^{3}$ Surface area of a sphere $=4 \pi R^{2}$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).(c) Assertion (A) is true and Reason (R) is false.(d) Assertion (A)...
Read More →The diagonals of a quadrilateral are of lengths 6 cm and 8 cm.
Question: The diagonals of a quadrilateral are of lengths 6 cm and 8 cm. If the diagonals bisect each other at right angles, what is the length of each side of the quadrilateral? Solution: Let the given quadrilateral be $\mathrm{ABCD}$ in which diagonals $\mathrm{AC}$ is equal to $6 \mathrm{~cm}$ and $\mathrm{BD}$ is equal to $8 \mathrm{~cm}$. Also, it is given that the diagonals bisect each other at right angle, at point $\mathrm{O}$. $\therefore \mathrm{AO}=\mathrm{OC}=\frac{1}{2} \mathrm{AC}=...
Read More →Assertion (A) If the radii of the circular ends of a bucket 24 cm high are 15 cm and 5 cm,
Question: Assertion (A)If the radii of the circular ends of a bucket 24 cm high are 15 cm and 5 cm, respectively, then the surface area of the bucket is 545 cm2.Reason(R)If the radii of the circular ends of the frustum of a cone areRandr,respectively, and its height ish, then its surface area is $\pi\left\{R^{2}+r^{2}+l(R-r)\right\}$, where $l^{2}=h^{2}+(R-r)^{2}$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Re...
Read More →Assertion (A) If the radii of the circular ends of a bucket 24 cm high are 15 cm and 5 cm,
Question: Assertion (A)If the radii of the circular ends of a bucket 24 cm high are 15 cm and 5 cm, respectively, then the surface area of the bucket is 545 cm2.Reason(R)If the radii of the circular ends of the frustum of a cone areRandr,respectively, and its height ish, then its surface area is $\pi\left\{R^{2}+r^{2}+l(R-r)\right\}$, where $l^{2}=h^{2}+(R-r)^{2}$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).(b) Both Assertion (A) and Re...
Read More →ABCD is a rhombus whose diagonals intersect at O.
Question: ABCDis a rhombus whose diagonals intersect atO. IfAB= 10 cm, diagonalBD= 16 cm, find the length of diagonalAC. Solution: We know that the diagonals of a rhombus bisect each other at right angles. $\therefore \mathrm{BO}=\frac{1}{2} \mathrm{BD}=\left(\frac{1}{2} \times 16\right) \mathrm{cm}$ $=8 \mathrm{~cm}$ $\mathrm{AB}=10 \mathrm{~cm}$ and $\angle \mathrm{AOB}=90^{\circ}$ From right $\Delta \mathrm{OAB}:$ $\mathrm{AB}^{2}=\mathrm{AO}^{2}+\mathrm{BO}^{2}$ $\Rightarrow \mathrm{AO}^{2}=...
Read More →Solve the following pairs of equations
Question: Solve the following pairs of equations (i) $x+y=3.3$, $\frac{0.6}{3 x-2 y}=-1,3 x-2 y \neq 0$ (ii) $\frac{x}{3}+\frac{y}{4}=4$, $\frac{5 x}{6}-\frac{y}{8}=4$ (iii) $4 x+\frac{6}{y}=15$, $6 x-\frac{8}{y}=14, y \neq 0$ (iv) $\frac{1}{2 x}-\frac{1}{y}=-1$ $\frac{1}{x}+\frac{1}{2 y}=8, x, y \neq 0$ (v) $43 x+67 y=-24$,$67 x+43 y=24$ (vi) $\frac{x}{a}+\frac{y}{b}=a+b$, $\frac{x}{a^{2}}+\frac{y}{b^{2}}=2, a, b \neq 0$ (vii) $\frac{2 x y}{x+y}=\frac{3}{2}$ $\frac{x y}{2 x-y}=\frac{-3}{10}, x+...
Read More →Match the following columns:
Question: Match the following columns: Solution: (a)Let R and r be the top and base of the bucket and let h be its height.Then, R = 20 cm, r = 10 cm and h = 30 cm.Capacity of the bucket = Volume of the frustum of the cone $=\frac{\pi h}{3}\left(R^{2}+r^{2}+R r\right)$ $=\frac{22}{7} \times \frac{1}{3} \times 30 \times\left[(20)^{2}+\left(10^{2}\right)+(20 \times 10)\right] \mathrm{cm}^{3}$ $=\frac{220}{7} \times[400+100+200] \mathrm{cm}^{3}$ $=\left(\frac{220}{7} \times 700\right) \mathrm{cm}^{3...
Read More →Show that each diagonal of a rhombus
Question: Show that each diagonal of a rhombus bisects the angle through which it passes. Solution: In $\triangle \mathrm{AED}$ and $\Delta \mathrm{DEC}:$ $\mathrm{AE}=\mathrm{EC}$ (diagonals bisect each other) $\mathrm{AD}=\mathrm{DC}$ (sides are equal) $\mathrm{DE}=\mathrm{DE}$ (common) $\mathrm{By} \mathrm{SSS}$ congruence : $\triangle \mathrm{AED} \cong \Delta \mathrm{CED}$ $\angle \mathrm{ADE}=\angle \mathrm{CDE}$ (c. p.c.t) Similarly, we can prove $\Delta \mathrm{AEB}$ and $\Delta \mathrm{...
Read More →Match the following columns:
Question: Match the following columns: Solution: (a)Let R and r be the top and base of the bucket and let h be its height.Then, R = 20 cm, r = 10 cm and h = 30 cm.Capacity of the bucket = Volume of the frustum of the cone $=\frac{\pi h}{3}\left(R^{2}+r^{2}+R r\right)$ $=\frac{22}{7} \times \frac{1}{3} \times 30 \times\left[(20)^{2}+\left(10^{2}\right)+(20 \times 10)\right] \mathrm{cm}^{3}$ $=\frac{220}{7} \times[400+100+200] \mathrm{cm}^{3}$ $=\left(\frac{220}{7} \times 700\right) \mathrm{cm}^{3...
Read More →ABCD is a rhombus and its diagonals intersect at O.
Question: ABCDis a rhombus and its diagonals intersect atO. (i) Is ∆BOC ∆DOC? State the congruence condition used? (ii) Also state, if BCO= DCO. Solution: (i) Yes In $\Delta \mathrm{BCO}$ and $\Delta \mathrm{DCO}:$ $\mathrm{OC}=\mathrm{OC}$ (common) $\mathrm{BC}=\mathrm{DC}$ (all sides of a rhombus are equal) $\mathrm{BO}=\mathrm{OD}$ (diagonal $s$ of a rhomus bisect each other) $\mathrm{By} \mathrm{SSS}$ congruence : $\Delta \mathrm{BCO} \cong \Delta \mathrm{DCO}$ (ii) Yes By c.p.c.t: $\angle B...
Read More →Draw a rhombus ABCD,
Question: Draw a rhombus ABCD, ifAB= 6 cm andAC= 5 cm. Solution: 1. Draw a line segment AC of 5 cm. 2. With A as centre, draw an arc of radius 6 cm on each side of AC. 3. With C as centre, draw an arc of radius 6 cm on each side of AC. These arcs intersect the arcs of step 2 at B and D. 4. Join AB, AD, CD and CB....
Read More →One side of a rhombus is of length 4 cm and the length of an altitude is 3.2 cm.
Question: One side of a rhombus is of length 4 cm and the length of an altitude is 3.2 cm. Draw the rhomb Solution: 1. Draw a line segment AB of 4 cm. 2. Draw a perpendicular XY on AB, which intersects AB at P. 3. With P as centre, cut PE at 3.2 cm. 4. Draw a line WZ that passes through E. This line should be parallel to AB. 5. With A as centre, draw an arc of radius 4 cm that cuts WZ at D. 6. With D as centre and radius 4 cm, cut line DZ. Label it as point C. 4. Join AD and CB....
Read More →Draw a rhombus, having each side of length 3.5 cm and one of the angles as 40°.
Question: Draw a rhombus, having each side of length 3.5 cm and one of the angles as 40. Solution: 1. Draw a line segment AB of 3.5 cm. 2. Draw $\angle B A X$ equal to $40^{\circ}$. 3. With A as centre and the radius equal to AB, cut AD at 3.5 cm. 4. With D as centre, cut an arc of radius 3.5 cm. 5. With B as centre, cut an arc of radius 3.5 cm. This arc cuts the arc of step 4 at C. 6. Join DC and BC....
Read More →Construct a rhombus whose diagonals are of length 10 cm and 6 cm.
Question: Construct a rhombus whose diagonals are of length 10 cm and 6 cm. Solution: 1. Draw AC equal to 10 cm. 2. Draw XY, the right bisector of AC, meeting it at O. 3. With O as centre and radius equal to half of the length of the other diagonal, i.e. 3 cm, cut OB = OD = 3 cm. 4. Join AB, AD and CB, CD....
Read More →If the diagonals of a rhombus are 12 cm and 16cm,
Question: If the diagonals of a rhombus are 12 cm and 16cm, find the length of each side. Solution: All sides of a rhombus are equal in length. The diagonals intersect at $90^{\circ}$ and the sides of the rhombus form right triangles. One leg of these right triangles is equal to $8 \mathrm{~cm}$ and the other is equal to $6 \mathrm{~cm}$. The sides of the triangle form the hypotenuse of these right triangles. So, we get: $\left(8^{2}+6^{2}\right) \mathrm{cm}^{2}$ $=(64+36) \mathrm{cm}^{2}$ $=100...
Read More →Match the following columns:
Question: Match the following columns: Solution: (a) Volume of the sphere $=\frac{4}{3} \pi r^{3}$ $=\left(\frac{4}{3} \pi \times(8)^{3}\right) \mathrm{cm}^{3}$ Volume of each cone $=\frac{1}{3} \pi r^{2} h$ $=\frac{1}{3} \pi \times(8)^{2} \times 4 \mathrm{~cm}^{3}$ Number of cones formed $=\frac{\text { Volume of the sphere }}{\text { Volume of each cone }}$ $=\frac{4 \pi \times 8 \times 8 \times 8 \times 3}{3 \times \pi \times 8 \times 8 \times 4}$ $=8$ Hence, $(a) \Rightarrow(q)$ (b) Volume o...
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