Exercise 6.5

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
$3 \mathrm{c}\mathrm{m}, 8 \mathrm{c}\mathrm{m}, 6 \mathrm{c}\mathrm{m}$

 

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
$50 \mathrm{c}\mathrm{m}, 80 \mathrm{c}\mathrm{m}, 100 \mathrm{c}\mathrm{m}$
 

The sides of a triangle are given below. Determine whether it is a right triangle. In case of a right triangle, write the length of its hypotenuse.
$13 \mathrm{c}\mathrm{m}, 12 \mathrm{c}\mathrm{m}, 5 \mathrm{c}\mathrm{m}$.

In the figure, $\mathrm{A}\mathrm{B}\mathrm{D}$ is a triangle right-angled at $\mathrm{}\mathrm{A}$ and $\mathrm{}\mathrm{A}\mathrm{C}\perp \mathrm{B}\mathrm{D}$. Show that  $\mathrm{A}{\mathrm{C}}^2=\mathrm{B}\mathrm{C}·\mathrm{D}\mathrm{C}$

In Fig. $\mathrm{A}\mathrm{B}\mathrm{D}$ is a triangle right-angled at $\mathrm{}\mathrm{A}$ and $\mathrm{}\mathrm{A}\mathrm{C}\perp \mathrm{B}\mathrm{D}$. Show that $\mathrm{A}{\mathrm{D}}^2=\mathrm{B}\mathrm{D}·\mathrm{C}\mathrm{D}$   

In Fig. $\mathrm{}\mathrm{O}$ is a point in the interior of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}, \mathrm{O}\mathrm{D}\perp \mathrm{B}\mathrm{C}, \mathrm{O}\mathrm{E}\perp \mathrm{A}\mathrm{C}$ and $\mathrm{}\mathrm{O}\mathrm{F}\perp \mathrm{A}\mathrm{B}$. Show that

 ${\mathrm{A}\mathrm{F}}^2+{\mathrm{B}\mathrm{D}}^2+{\mathrm{C}\mathrm{E}}^2={\mathrm{A}\mathrm{E}}^2+{\mathrm{C}\mathrm{D}}^2+{\mathrm{B}\mathrm{F}}^2.$

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

In $∆\mathrm{A}\mathrm{B}\mathrm{C}, \mathrm{A}\mathrm{B}=6\sqrt{3} \mathrm{c}\mathrm{m}, \mathrm{A}\mathrm{C}=12 \mathrm{c}\mathrm{m}$ and $\mathrm{B}\mathrm{C}=6 \mathrm{c}\mathrm{m}$. The angle $\mathrm{}\mathrm{B}$ is