Internal and External Bisector of an Angle

In $\text{△ABC}$,$\mathrm{AD}$ is the bisector of $\angle \mathrm{A}$  that intersects BC in D. If $\mathrm{AB} = 8 \mathrm{cm}$ and $\mathrm{AC} = 12 \mathrm{cm},$ then find BD : DC?

In the adjoining figure if exterior $\angle EAB= 110°, \angle CAD= 35°, AB= 5 \mathrm{cm}, AC= 7 \mathrm{cm}\text{ and} BC= 3 \mathrm{cm}$, then $CD$ is equal to:

In the figure, Point $Q$ is on the side of $MP$ such that $MQ=2$ and $MP=5.5$. Ray $NQ$ is the bisector of $\angle MNP$ of $∆MNP$. If $MN:NP$ is $k:1$,then find $k+1$.

In what ratio of line $x-y-2=0$ divides the line segment joining $\left(3,-1\right)$ and $\left(8,9\right)$?

In the given fig.$AM\perp BC$ and $AN$ is the bisector of $\angle A$. If $\angle B=65°$ and $\angle C=33°$, then the value of $\angle MAN$ will be

In the given figure, $AD$ is the bisector of $\angle BAC$. If $AB=10cm, AC=6cm$ and $BC=12cm$. Find $BD$.

In $∆\mathrm{ABC}$, $\angle \mathrm{ABC}=90°$ and$\angle \mathrm{BAC}=60°$. If bisector of $\angle \mathrm{BAC}$ meets $\mathrm{BC}$ at $\mathrm{D}$. Then $\mathrm{BD}:\mathrm{DC}$ is

$D$ and $E$ are the points on sides $BC$ and $AC$, respectively of $∆ABC. AD$ and $BE$ intersect each other at T. If $\frac{AT}{TD}=5$ and $\frac{BT}{ET}=7$ , then $CD : BD =$

In the figure above, $AD\perp AB, CD\perp BC, \angle ABD=x-23°, \angle DBC=1°-x$ and $AD=CD$. Find $x$.

In the figure above, $AD\perp AB, CD\perp BC, \angle ABD=25°$ and $AD=CD$. Find $\angle DBC$.

In the figure above, $AD\perp AB, CD\perp BC, \angle DBC=30°$ and $AD=CD$. Find $\angle ABD$.

In  the given figure, $G$ is the centroid of $∆DEF$ and point I is the intersection point of its three angle bisectors. $DE=8, DF=12$ and $EF=10$ are given. Find the length of $IG.$

In the given figure, G is the centroid of $∆ABC$ and $AE=EC.$If $FC=18,$ What is the length of the $GK?$

In the given figure, $G$ is the centroid of $∆ABC, BD=DC$ and $GD=\frac{5}{2}.$ Find the length of $BC.$

Find the value of $m^2+n^2$ by using the information given in figure.

In the given figure, $A{C}^{\text{'}}=B{C}^{\text{'}}, B{A}^{\text{'}}=C{A}^{\text{'}}, C^{\text{'}}D=2. DC$ and $A{A}^{\text{'}}\perp C{C}^{\text{'}}.$ If $A^{\text{'}}D=4,$ what is the length of the $AC?$

In the figure, $G$ is the centroid $∆ABC.$ If $AE\perp BC, AG=12$ and $DG=5,$ find the length of $BC.$

In an equilateral triangle $ABC$ of side $28 cm$. $\frac{AN}{NB}=\frac{BL}{LC}=\frac{CM}{MA}=\frac{2}{1}$

Then the area of triangle of $△PQR$ is 

From the diagram find the area of the $△AED$

In $△ABC$ area of region (in sq. units) are mentioned in the figure. Find the area of the quadrilateral $AEFD$.