Pythagoras Theorem

In a $△ABC$, $D$ is the mid-point of $BC$ and $E$ is the foot of the perpendicular drawn from $A$ on $BC$. If $ED=x , AD=p, AE=h$ and $BC=a$, then $2b^2+2c^2-a^2=$_____

$ABCD$ is a rectangle. Taking $AD$  as diameter a semi-circle is drawn which cuts the diagonal $DB$ at $E$. If $AB=12 \mathrm{cm}$ and $AD=9 \mathrm{cm}$, then $BE=$_____$\mathrm{cm}$.

Two circles of radii $9 \mathrm{cm} \text{and} 25 \mathrm{cm}$ touch each other externally. The length of a direct common tangent is_____$\mathrm{cm}$.

$ABCD$ is a rectangle such that $AC+AB=5AD$ and $AC–AD=8$, then the area of rectangle $ABCD$ is

In a $∆ABC,\angle C=90°$.  On the sides $CA$ and $CB$ two points $P$ and $Q$ are taken such that they divide $CA$ and $CB$ in the ratio $2:1$ respectively. Then, $\left(A{Q}^2+B{P}^2\right):A{B}^2=$

If in a triangle, square of longest side is equal to the sum of the squares of the other two sides, then the angle opposite the longest side is

In $△ABC$, $\angle 90°$ and points $D$ and $E$ divide $BC$ into three equal parts. If $3{AC}^2+5{AB}^2=p{AE}^2$ then $p=$_____

In a right angled triangle, the difference of the hypotenuse and the base is $2 \mathrm{cm}$. the hypotenuse exceeds twice the height by $1 \mathrm{cm}$. The base of the triangle is_____$\mathrm{cm}$

For going to a city $B$ from city $A$, there is a route via city $C$ such that $AC\perp CB$, $AC=2x \mathrm{km}$ and $CB=2\left(x+7\right) \mathrm{km}$. It is proposed to construct a $26 \mathrm{km}$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway.

$PQRS$ is a square of side $6 \mathrm{cm}$ each and $T$ is mid-point of $QR$. What is the radius of circle inscribed in $△TSR$ .

Three circles touch each other externally and all the three touch a line. If two of them are equal and radius of third circle is $4 \mathrm{cm}$ then radius of equal circles is:

In the given figure $OCDE$ is a rectangle inscribed in a quadrant of a circle of$10 \mathrm{cm}$ radius. If $OE=2\sqrt{5} \mathrm{cm}$, then the area of rectangle will be-

If each side of a Rhombus is $20cm$ and its shorter diagonal is $\frac{3}{4}$ of its longer diagonal then the area of Rhombus will be _____.

$\mathrm{ABC}$ is a triangle in which $\mathrm{AB}=10\mathrm{cm}, \mathrm{AC}=24\mathrm{cm}$ and $\mathrm{BC}=26\mathrm{cm}$. If $\mathrm{AD}$ is its median then length of $\mathrm{AD}$ is

$\mathrm{ABC}$ is a right-angled triangle, right-angled at $\angle \mathrm{B}$. If side $\mathrm{AB}$ is divided into three equal parts by points $\mathrm{D}$ and $\mathrm{E}$ such that $\mathrm{D}$ is nearer to $\mathrm{A}$, then $\frac{A{C}^2-E{C}^2}{D{C}^2-B{C}^2}$ is equal to

Two concentric circles with centre $O$, have radii $15 \mathrm{cm} \text{and} 9 \text{cm}$. From a point $A$ on the bigger circle tangents $AB \text{and} AC$ are drawn to the smaller circle at $B \text{and} C$, respectively, intersecting bigger circles at $D \text{and} E,$ respectively, $OF\perp DE$ at $F$. The length of $OF$ is:

In the scalene $∆ABC$, the smallest side is $28 m$, the largest side is $100 m$, and the smallest angle equals the difference between the larger two angles. Find the length of the third side in metres.

The medians of a right-angled triangle which are drawn from the vertices of the acute angles are $6 cm$ and $8 cm$, then find the length of the hypotenuse.

$∆ABC$ is right-angled at $B$. $BD$ is perpendicular upon $AC$. If $AD=a$ and $CD=b$, then find the value of $A{B}^2$.

In the figure, triangle ABC is a right-angled triangle, and DEHF is a square. If AH=3 and BD=8, what is the length of one side of DEFH?.