Pythagoras Theorem

For a right angle triangle with integer sides at least one of its measurements must be an even number. Why?

In a right triangle $ABC$ right-angled at $C$. $P$ and $Q$ are points on sides $AC$ and $CB$ respectively which divide these sides in the ratio of $2:1$.

Prove that $9(A{Q}^2+B{P}^2)=13A{B}^2$.

In a right triangle $ABC$ right-angled at $C$. $P$ and $Q$ are points on sides $AC$ and $CB$ respectively which divide these sides in the ratio of $2:1$.

Prove that $9B{P}^2=9B{C}^2+4A{C}^2$.

In a right triangle $ABC$ right-angled at $C$. $P$ and $Q$ are points on sides $AC$ and $CB$ respectively which divide these sides in the ratio of $2:1$.

Prove that $9A{Q}^2=9A{C}^2+4B{C}^2$

An aeroplane leaves an airport and flies due north at a speed of $1000 \mathrm{kmph}$. At the same time another aeroplane leaves the same airport and flies due west at a speed of $1200 \mathrm{kmph}$. How far apart will the two planes be after $1\frac{1}{2}$ hours?

$ABC$ is an isosceles triangle right angle at $C$. Prove that $A{B}^2=2A{C}^2$.

$ABD$ is a triangle right angle at $A$ and $AC\perp BD$. Show that $A{B}^2=BC\times BD$.

$ABD$ is a triangle right angle at $A$ and $AC\perp BD$. Show that $A{C}^2=BC\times DC$.

$ABD$ is a triangle right angle at $A$ and $AC\perp BD$. Show that $A{D}^2=BD\times CD$.

$PQR$ is a triangle right angle at $P$ and $M$ is a point on $QR$ such that $PM\perp QR$. Show that $P{M}^2=QM\times MR$.

$ABC$ is an isosceles triangle right-angled at $B$. Equilateral triangles $ACD$ and $ABE$ are constructed on sides $AC$ and $AB$. Find the ratio between the areas of $∆ABE$ and $∆ACD$.

In the given figure, $ABC$ is a triangle right-angled at $B$. $D$ and $E$ are points on $BC$ trisect it. Prove that $8A{E}^2=3A{C}^2+5A{D}^2$.

In an equilateral triangle$ABC$, $D$ is a point on side $BC$ such that $BD=\frac{1}{3}BC$. Prove that $9A{D}^2 =7A{B}^2$.

Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.

Two poles of heights $6 \mathrm{m}$ and $11 \mathrm{m}$ stand on a plane ground. If the distance between the feet of the poles is $12 \mathrm{m}$, find the distance between their tops in meters.

$ABC$ is aright triangle right-angled at $B$. Let $D$ and $E$ be any points on $AB$ and $BC$ respectively. Prove that $A{E}^2+C{D}^2=A{C}^2+D{E}^2$.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

A wire attached to vertical pole of height $18 \mathrm{m}$ is $24 \mathrm{m}$ long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

In figure, $O$ is any point in the interior of $∆ABC$. If $OD\perp BC,OE\perp AC$ and $OF\perp AB$, show that $A{F}^2+B{D}^2+C{E}^2=A{E}^2+C{D}^2+B{F}^2$.

In the given figure, $AD\perp BC$. Prove that $A{B}^2+C{D}^2=B{D}^2+A{C}^2$.