Chapterwise Revision-(Stage 3)

The following figure shows a triangle $ABC$ in which $AD$ is a median and $AE\perp BC$ . Prove that : $2A{B}^2+2A{C}^2=4A{D}^2+B{C}^2$

Construct a quadrilateral $ABCD$ in which $AB=4.5 \mathrm{cm}, BC=3.8 \mathrm{cm}, \angle BCD=90°, \angle BAD=60°$ and $\angle ABC=120°$.

Alternate sides of a hexagon are produced to meet so as to form a star-shaped figure. Show that the sum of the angles at vertices of the star is equal to four-right angles.

In a parallelogram $ABCD, AB=\left(3x-4\right) \mathrm{cm}, BC=\left(y-1\right) \mathrm{cm}, CD=\left(y+5\right) \mathrm{cm}$ and $AD=\left(2x+5\right) \mathrm{cm}$, find the ratio $AB:BC$. Find the values of $x$ and $y$.

Construct quadrilateral $ABCD$ in which $BC=5 \mathrm{cm}, \angle ABC=60°, AB-AC=1.5 \mathrm{cm}$ and $△ADC$ is equilateral.

In the given figure $AB=AC, \angle A = \angle D = 90°, BD= 10 cm and DC = 24 cm,$Calculate the length of AB correct to two place decimal. Also , find the perimeter of quadrilateral $ABCD$

In a right angled triangle, five times the square on the hypotenuse is equal to four times the sum of the squares on the medians drawn from the acute angle. Prove it.

In an octagon, four of the angles are equal and each other angle is $20°$ more than each of the first four. Find the angles of the octagon.

Each interior angle of a regular polygon is $144°$. Find the interior angle of a regular polygon which has double the number of sides as the first polygon.

Find the numbers of sides of a polygon whose number of diagonals is $5$.

Find the numbers of sides of a polygon whose number of diagonals is $14$.

Find the numbers of sides of a polygon whose number of diagonals is $27$.

In the $△ABC, \angle ABC=90°, AB=2a+1$ and $BC=2a^2+2a$. Find $AC$ in the terms of $\text{'}a\text{'}.$ If $a=8 \mathrm{cm}$ find the lengths of the sides of the $△ABC.$

If $x=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and $y=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$, find the value of $x^2+y^2$.

In triangle $\mathrm{ABC}, \mathrm{AD}$ is perpendicular to $\mathrm{BC}$ and ${\mathrm{AD}}^2=\mathrm{BD}\times \mathrm{DC}.$ Show that angle $\mathrm{BAC}=90º.$

A river $1 \mathrm{m}$ deep and $40 \mathrm{m}$ wide is flowing at the rate of $2 \mathrm{km}/\mathrm{h}$. How much water will fall into the sea in a $\mathrm{minute}$?

A small indoor green house(herbarium) is made entirely of glass panes including base, held together with tape. It is $30 \mathrm{cm}$ long, $25 \mathrm{cm}$ wide and $25 \mathrm{cm}$ height. How much tape is needed for all the $12$ edges?

A small indoor green house(herbarium) is made entirely of glass panes including base, held together with tape. It is $30 \mathrm{cm}$ long, $25 \mathrm{cm}$ wide and $25 \mathrm{cm}$ height. What is the area of the glass used?

The internal dimensions of a box are $1.2 \mathrm{m}, 80 \mathrm{cm}$ and $50 \mathrm{cm}$. How many cubes each of edge $7 \mathrm{cm}$ can be packed in the box with faces parallel to the sides of the box. Also, find the space left empty in the box.

A wall $24 \mathrm{m}$ long, $5 \mathrm{m}$ high and $0.25 \mathrm{m}$ thick is to be constructed using bricks each measuring $25 \mathrm{cm}\times 12.5 \mathrm{cm}\times 7.5 \mathrm{cm}$. Find the number of bricks required, if $5\%$ of the wall is occupied by cement and sand mixture.