Introduction to Pythagoras Theorem

In $△\mathrm{ABC}$, $\angle \mathrm{B}=90°$ and $\mathrm{D}$ is the mid point of $\mathrm{BC}$, Prove that: ${\mathrm{AC}}^2={\mathrm{AD}}^2+3{\mathrm{CD}}^2$

In quadrilateral $\mathrm{ABCD}, \angle \mathrm{B}=\angle \mathrm{D}={90}^{°}.$ Prove that $2{\mathrm{AC}}^2-{\mathrm{BC}}^2={\mathrm{AB}}^2+{\mathrm{AD}}^2+{\mathrm{DC}}^2$.

In triangle $\mathrm{ABC}, \mathrm{AD}\perp \mathrm{BC}$, prove that $\left(\mathrm{AB}+\mathrm{AC}\right)\left(\mathrm{AC}-\mathrm{AB}\right)=\left(\mathrm{CD}+\mathrm{BD}\right)\left(\mathrm{CD}-\mathrm{BD}\right)$.

In $△\mathrm{PQR}, \mathrm{PS}$ is perpendicular to $\mathrm{RQ}$ extended. Prove that ${\mathrm{PR}}^2={\mathrm{PQ}}^2+{\mathrm{QR}}^2+2\mathrm{QR}.\mathrm{SQ}$.

In $△\mathrm{ABC}, \mathrm{AB}=8$, find a and b if $\mathrm{a}+\mathrm{b}=32$.

$\mathrm{P}$ is any point inside the rectangle $\mathrm{ABCD}.$Prove that; ${\mathrm{PA}}^2+{\mathrm{PC}}^2={\mathrm{PB}}^2+{\mathrm{PD}}^2$.

If $PQ=2m, QR=m^2-1 \& PR=m^2+1$, show that $PQR$ is a right-angled triangle. Hence, find the sides of the triangle when $m=4, 5, 6.$

In $△ABC,\angle B=90°,AP=5, PB=15, PQ=17 \& QC=7$. Find the value of $x \& y$.

All the measurements are in centimeters.

For a standard television set, the horizontal length to the height is in the ratio $4:3$. Find the length of $35$ inches television.

(The TV measurement indicates its diagonal length).

$AB \& CD$ are two buildings of height $30 m \& 63 m$. If the distance between them is $56 m,$ calculate the distance between their top.

In the quadrilateral $ABCD, \angle B=90°=\angle D, AB=7 cm, BC=17 cm \& AD=DC=x$. Find the value of $x$.

In $△ABC, AD\perp BC. D$ divides $BC$ in the ratio $2:3$. Prove that $5A{C}^2=5A{B}^2+B{C}^2$.

In $△ABC, \angle BAC=90°=\angle APB. BC=85 cm, AC=75 cm \& BP=32 cm.$ Find $x \& y$.

$ABCD$ is a rectangle. $AP=6 cm, PB=5 cm, AQ=8 cm, QD=4 cm, DR=3 cm$. FInd the $PQ, QR \& PC.$

$PQRS$ is a rectangle. $PA=16 cm, AQ=9 cm \& QR=12 cm$. Prove that $\angle SAR=90°$.

$PQRS$ is a rectangle. $PA=16 cm, AQ=9 cm \& QR=12 cm$. Find the lengths of $AS \& AR$.

The upper part of a tree is broken by the wind and falls on the ground without being detached. If the top of the broken part touches the ground at a distance of $12$ feet from the foot of the tree, calculate the height at which it is broken if the original height is $24$ feet.

In the rectangle $ABCD, AB=12 cm, BC=7 cm. P \& R$ are mid-points of $AB \& CD. AS=BQ=2.5 cm.$  Find the perimeter of $PQRS$.

$PQRS$ is a rectangle. Prove that $P{R}^2+Q{S}^2=P{Q}^2+Q{R}^2+R{S}^2+P{S}^2$.

In $△ABC, \angle B=90°, AB=BC. BC$ is produced to a point $D$. Prove that $A{D}^2=A{C}^2+C{D}^2+2.BC.CD$.