Pythagoras Theorem

Prove pyathagoras theorem

In a right angle $∆ABC$, $\angle A$  is a right angle. $P$ and $Q$ are any points on $AB$ and $AC$. Prove that, $B{Q}^2+P{C}^2=B{C}^2+P{Q}^2$.

In the given figure, $\mathrm{D}$ is the midpoint of side$\mathrm{BC}$ and $\mathrm{AE}\perp \mathrm{BC}$. If $\mathrm{BC}=\mathrm{a}, \mathrm{AC}=\mathrm{b}, \mathrm{AB}=\mathrm{c} , \mathrm{ED}=\mathrm{x}, \mathrm{AD}=\mathrm{p}$ and $\mathrm{AE}=\mathrm{h}$, prove that ${\mathrm{c}}^2={\mathrm{p}}^2-\mathrm{ax}+\frac{{\mathrm{a}}^2}{4}$.
 

In a given figure, $\mathrm{D}$ is the midpoint of side$\mathrm{BC}$ and $\mathrm{AE}\perp \mathrm{BC}$. If $\mathrm{BC}=\mathrm{a}, \mathrm{AC}=\mathrm{b}, \mathrm{AB}=\mathrm{c} , \mathrm{ED}=\mathrm{x}, \mathrm{AD}=\mathrm{p}$ and $\mathrm{AE}=\mathrm{h}$, by using Pythagoras theorem prove that ${\mathrm{b}}^2+{\mathrm{c}}^2=2{\mathrm{p}}^2+\frac{{\mathrm{a}}^2}{2}$.

Two trains leave a railway station at the same time. The first train travels due west and the $2^{nd}$ train due north. The first train travels at a speed of $20 \mathrm{km}/\mathrm{hr}$ and the $2^{nd}$ train travels at $30 \mathrm{km}/\mathrm{hr}$. After two hours, let the distance between them is $\mathrm{k}\sqrt{13} \mathrm{km}$. Find the value of $\mathrm{k}$.

A right-angled triangle $ABC$, $\angle ABC=90°, AB=3 \mathrm{cm}, BC=$four $cm$ and the perpendicular $BD$ on the side $AC$ from the point $B$ which meets the side $AC$ at the point $D$. Determine the length of $BD$.

Identify the reason, why it is Pythagoras triplet :

$\left(3,5,4\right)$

Identify the reason, why it is not Pythagoras triplet :

$\left(4,9,12\right)$

In $∆\mathrm{ABC}, \angle \mathrm{BAC}$ one right angle. $\mathrm{AD}$ is perpendicular to $\mathrm{BC}$. We need to prove that ${\mathrm{AD}}^2=\mathrm{BD}.\mathrm{CD}$.

In $∆\mathrm{ABC},$the perpendicular from $\mathrm{A}$ upon $\mathrm{BC}$ intersects $\mathrm{BC}$ at $\mathrm{D}$. If $\mathrm{BD}=16 \mathrm{cm}, \mathrm{DC}= 4 \mathrm{cm}$ and $\mathrm{AD}=8 \mathrm{cm}$, then find $\angle \mathrm{BAC}?$

In $∆\mathrm{ABC},$the perpendicular from $\mathrm{A}$ upon $\mathrm{BC}$ intersects $\mathrm{BC}$ at $\mathrm{D}$. If $\mathrm{BD}=4 \mathrm{cm}, \mathrm{DC}= 1 \mathrm{cm}$ and $\mathrm{AD}=2 \mathrm{cm}$, then find $\angle \mathrm{BAC}?$

In $∆\mathrm{ABC},$the perpendicular from $\mathrm{A}$ upon $\mathrm{BC}$ intersects $\mathrm{BC}$ at $\mathrm{D}$. If $\mathrm{BD}=9 \mathrm{cm}, \mathrm{DC}= 3 \mathrm{cm}$ and $\mathrm{AD}=6 \mathrm{cm}$, then find $\angle \mathrm{BAC}?$

In $∆\mathrm{ABC},$the perpendicular from $\mathrm{A}$ upon $\mathrm{BC}$ intersects $\mathrm{BC}$ at $\mathrm{D}$. If $\mathrm{BD}=8 \mathrm{cm}, \mathrm{DC}= 2 \mathrm{cm}$ and $\mathrm{AD}=4 \mathrm{cm}$, then find $\angle \mathrm{BAC}?$

In right-angled triangle $\mathrm{ABC}$, $\angle \mathrm{ABC}=$one right angle and $\mathrm{AB}=2 \mathrm{cm}$, $\mathrm{BC}=4 \mathrm{cm}$. What is the circumradius of $∆\mathrm{ABC}$?

In right-angled triangle $\mathrm{ABC}$, $\angle \mathrm{ABC}=$one right angle and $\mathrm{AB}=5 \mathrm{cm}$, $\mathrm{BC}=6 \mathrm{cm}$. What is the circumradius of $∆\mathrm{ABC}$?

In right-angled triangle $\mathrm{ABC}$, $\angle \mathrm{ABC}=$one right angle and $\mathrm{AB}=3 \mathrm{cm}$, $\mathrm{BC}=12 \mathrm{cm}$. What is the circumradius of $∆\mathrm{ABC}$?

In right-angled triangle $\mathrm{ABC}$, $\angle \mathrm{ABC}=$one right angle and $\mathrm{AB}=5 \mathrm{cm}$, $\mathrm{BC}=12 \mathrm{cm}$. What is the circumradius of $∆\mathrm{ABC}$?

In a right angle triangle, the length of the sides containing the right angle are $\left(x-3\right) \mathrm{cm}$ and $2\sqrt{3x} \mathrm{cm}$. Then, the length of the hypotenuse is $\left(x+3\right) \mathrm{cm}$.

Out of the two concentric circles, the diameter of the outer circle is $10 \mathrm{cm}$ and the chord $\mathrm{AC}$ of length $8 \mathrm{cm}$ is a tangent to the inner circle, then find the diameter(in cm) of the inner circle. (Write the final answer without unit)

The diagonals of a rhombus are $30 \mathrm{cm}$ and $40 \mathrm{cm}.$ If each side of the rhombus is $l \mathrm{cm}$, then find the value of $l$.