Manipur Board Solutions for Chapter: Triangles, Exercise 4: EXERCISE 7.4

$ABC$ is a right triangle right-angled at $B$ and $D$ is the foot of the perpendicular drawn from $B$ on $AC$. If $DM\perp BC$ and $DN\perp AB$ where $M, N$ lie on $BC$, $AB$ respectively, prove that $D{M}^2=DN\times MC$.

$ABC$ is a right triangle right-angled at $B$ and $D$ is the foot of the perpendicular drawn from $B$ on $AC$. If $DM\perp BC$ and $DN\perp AB$ where $M, N$ lie on $BC$, $AB$ respectively, prove that $D{N}^2=DM\times AN$.

Through the vertex $D$ of parallelogram $ABCD$, a line is drawn to intersects the sides $BA$ produced and $BC$ produced at $E$ and $F$ respectively. Prove that $\frac{AD}{AE}=\frac{BF}{BE}=\frac{CF}{CD}$

The perimeter of two similar triangle $ABC$ and $PQR$ respectively, $72 cm$ and $48 cm$. If $PQ=20 cm$, find $AB$

Two right triangles $ABC$ and $DBC$ are drawn on the same hypotenuse $BC$ and on the same side of $BC$. If $AC$ and  $BD$ intersects at P$P$, prove that $AP\times PC=PD\times PB$ 

If one angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite sides in the same ratio, prove that the triangles are similar.

If two triangles are equiangular, prove that the ratio of the corresponding sides is same as the ratio of the corresponding altitudes.

If two triangles are equiangular , prove that the ratio of the corresponding sides is same as the ratio of the corresponding angle bisector segments.