R Chandra Solutions for Chapter: Triangles, Exercise 4: TEST YOUR SKILLS

$\mathrm{D}$ is a point on side $\mathrm{B}\mathrm{C}$ of $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ such that $\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{C}$ (see figure). Show that $\mathrm{A}\mathrm{B}>\mathrm{A}\mathrm{D}$.

In the figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ is a triangle such that $\mathrm{\angle }\mathrm{B}=40°$ and $\mathrm{\angle }\mathrm{C}=55°$ . The bisector of $\mathrm{\angle }\mathrm{A}$ meets $\mathrm{B}\mathrm{C}$ at $\mathrm{X}$. Show that $\mathrm{B}\mathrm{X}>\mathrm{A}\mathrm{X}>\mathrm{C}\mathrm{X}$.

Show that the difference of any two sides of a triangle is less than the third side.

Prove that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle.

In the figure, $\mathrm{P}\mathrm{Q}=\mathrm{P}\mathrm{R}$. If $\mathrm{S}$ is a point on $\mathrm{P}\mathrm{R}$, then prove that $\mathrm{R}\mathrm{S}<\mathrm{Q}\mathrm{S}$.

Show that the perimeter of a triangle is greater than the sum of its three medians.

In the figure, $\mathrm{AD}\perp \mathrm{B}\mathrm{C},\hspace{0.17em}\hspace{0.17em}\mathrm{D}\mathrm{C}>\mathrm{D}\mathrm{B}$ and $\mathrm{D}\mathrm{E}=\mathrm{D}\mathrm{B}$. Show that $\mathrm{A}\mathrm{C}>\mathrm{A}\mathrm{B}$.

In the given figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{A}\mathrm{B}$ is a diameter. If $\mathrm{A}\mathrm{C}$ is any other chord of the circle, then show that $\mathrm{A}\mathrm{B}>\mathrm{A}\mathrm{C}$.