Embibe Experts Solutions for Chapter: Triangles, Exercise 1: Jammu and Kashmir Board-2018

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Prove it.

If $\mathrm{}\mathrm{A}\mathrm{D}$ and $\mathrm{}\mathrm{P}\mathrm{M}$ are medians of triangles $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{}\mathrm{P}\mathrm{Q}\mathrm{R}$, respectively where $\mathrm{}∆\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{}~\mathrm{}∆\mathrm{}\mathrm{P}\mathrm{Q}\mathrm{R}$, prove that $\frac{\mathrm{A}\mathrm{B}}{\mathrm{P}\mathrm{Q}}=\frac{\mathrm{A}\mathrm{D}}{\mathrm{P}\mathrm{M}}$.

The altitude of a right triangle is $7 \mathrm{cm}$ less than its base. If the hypotenuse is $13 \mathrm{cm}$, find the other two sides.

The diagonal of a rectangular field is $60$ meters more than its shorter side. If the longer side is $30$ meters more than the shorter side. Find the sides of the field.

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

$ABCD$ is a trapezium in which $AB\parallel DC$ and its diagonal intersect each other at a point $O$. Show that:

$\frac{AO}{BO}=\frac{CO}{OD}$

$BL$ and $CM$ are the medians of a triangle $ABC$, right angled at $A$. Prove that: $4\left(B{L}^2+C{M}^2\right)=5B{C}^2$

$D$ and $E$ are the points on the sides $CA$ and $CB$ respectively of a triangle $ABC$ right angled at $C$. Prove that, $A{E}^2+B{D}^2=A{B}^2+D{E}^2$