Manipur Board Solutions for Chapter: Triangles, Exercise 6: EXERCISE 7.6

Find the length of the altitude and area of an equilateral triangle having $\text{'}a\text{'}$ as length of a side

If $D, E, F$ are the mid-points of the sides $BC, CA, AB$ of a right $△ABC$ (right-angled at $A$) respectively, prove that $3(A{B}^2+B{C}^2+C{A}^2)=4(A{D}^2+B{E}^2+C{F}^2)$

In a right triangle $ABC$ right angled at $C$, if $p$ is the length of the perpendicular segment drawn from $C$ upon $AB$, then prove that  $ab=pc$ 

In a right triangle $ABC$ right angled at $C$, if $p$ is the length of the perpendicular segment drawn from $C$ upon $AB$, then prove that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}$ where $a=BC, b=CA$ and $c=AB.$

In an equilateral triangle $ABC$ the side $BC$ is trisected at $D$. Prove that $9A{D}^2=7A{B}^2$

If $A$be the area of a right triangle and $b$one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is $\frac{2Ab}{\sqrt{b^4}+4A^2}$

Prove that the equilateral triangles described on the two sides of a right-angled triangle are together equal to the equilateral triangle on the hypotenuse in terms of their area.

If $O$ is any point in the interior of a rectangle $ABCD$, prove that $O{A}^2+O{C}^2=O{B}^2+O{D}^2$