Embibe Experts Solutions for Chapter: Triangles, Exercise 1: Nagaland Board-2019

State and prove Basic Proportionality Theorem.

$OA, OB$ and $OC$ are three lines segments. $L, M, N$ are points on $OA, OB$ and $OC$  respectively such that $LM\parallel AB, MN\parallel BC$. Prove that $LN\parallel AC$.

In the adjoining figure, if $PQ\parallel BC$, Find $x$.

In the adjoining figure, $PA, QB$ and $RC$ are perpendicular to $AC$ such that $PA=x, RC=y, QB=z, AB=a$ and $BC=b$. Prove that $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$.

State and prove Pythagoras theorem.

$ABCD$ is a trapezium in which $AB\parallel DC$, $AB=2DC$ and the diagonals intersect each other at $O$. If $ar\left(∆AOB\right)=84 {\mathrm{cm}}^2$, find $ar\left(∆COD\right)$.

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

In the adjoining figure, the perpendicular from $A$ on side $BC$ of a $∆ABC$ intersects $BC$ at $D$ such that $DB=3CD$. Prove that $2A{B}^2=2A{C}^2+B{C}^2$