NCERT Solutions for Chapter: Triangles, Exercise 4: Exercise

In a right triangle, prove that the line-segment joining the midpoint of the hypotenuse to the opposite vertex is half the hypotenuse.

Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through the point $O$ such that $P$ is equidistant from $l$ and $m$. Prove that $n$ is the bisector of the angle formed by $l$ and $m$.

Line-segment joining the midpoints $M$ and $N$ of parallel sides $AB$ and $DC$ respectively of trapezium $ABCD$ is perpendicular to both the sides $AB$ and $DC$. Prove that $\mathrm{AD}=\mathrm{BC}$.

$ABCD$ is a quadrilateral such that diagonal $AC$ bisects the angles $A$ and $C$. Prove that $\mathrm{AB}=\mathrm{AD}$ and $\mathrm{CB}=\mathrm{CD}$.

$ABC$ is a right triangle such that $AB=AC$ and bisector of angle $C$ intersects the side $AB$ at $D$. Prove that $AC+AD=BC$.

$AB$ and $CD$ are the smallest and largest sides of a quadrilateral $ABCD$. Out of $\angle \mathrm{B}$ and $\angle \mathrm{D}$ decide which is greater.

Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than $\frac{2}{3}$ of a right angle.

$ABCD$ is quadrilateral such that $\mathrm{AB}=\mathrm{AD}$ and $\mathrm{CB}=\mathrm{CD}.$ Prove that $\mathrm{AC}$ is the perpendicular bisector of $BD$.