Thales Theorem and Angle Bisector Theorem

In the given figure, $AD$ is the bisector of $\angle BAC$, if $AB=10 \mathrm{cm}, AC=14 \mathrm{cm}$ and $BC=6 \mathrm{cm}$. Find $BD$ and $DC$.

In the given figure, $AD$ is the bisector of $\angle A$. If $BD=4 \mathrm{cm}, DC=3 \mathrm{cm}$ and $AB=6 \mathrm{cm}$, find $AC$.

Draw a triangle $ABC$ of base $BC=8 \mathrm{cm}$, $\angle A={60}^{\mathrm{o}}$ and the bisector of $\angle A$ meets $BC$ at $D$ such that $BD=6 \mathrm{cm}$

Construct a triangle $△PQR$ such that $QR=5 \mathrm{cm}, \angle P={30}^{\mathrm{o}}$ and the altitude from $P$ to $QR$ is of length $4.2 \mathrm{cm}$.

Construct a $△PQR$ in which $PQ=8 \mathrm{cm}, \angle R={60}^{\mathrm{o}}$ and the median $RG$ from $R$ to $PQ$ is $5.8 \mathrm{cm}$. Find the length of the altitude from $R$ to $PQ$.

In $∆ABC$, if $DE\parallel BC, AD=x, DB=x-2$ and $EC=x-1$ then find the lengths of the sides $AB$ and $AC$.

If the median $AD$ to the side $BC$ of a $△ABC$ is also an angle bisector of $\angle A$ then $\frac{AB}{AC}$ is _____.

The _____ of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle.

Let $△ABC$ be equilateral. If $D$ is a point on $BC$ and $AD$ is the internal bisector of $\angle A$. Using Angle Bisector Theorem, $\frac{BD}{DC}$ is_____.

Basic Proportionality Theorem is also known as _____.

A straight line drawn _____ to a side of a triangle divides the other two sides proportionally.

An _____ is a cevian that divides the angle, into two equal halves.

Check whether $\mathrm{AD}$ is bisector of $\angle \mathrm{A}$ of $△\mathrm{ABC}$ in the following,

$\mathrm{AB}=4 \mathrm{cm}, \mathrm{AC}=6 \mathrm{cm}, \mathrm{BD}=1.6 \mathrm{cm}$ and $\mathrm{CD}=2.4 \mathrm{cm}$.

Write the answer as Yes / No.

Check whether $\mathrm{AD}$ is bisector of $\angle \mathrm{A}$ of $△\mathrm{ABC}$ in the following,

$\mathrm{AB}=5 \mathrm{cm}, \mathrm{AC}=10 \mathrm{cm}, \mathrm{BD}=1.5 \mathrm{cm}$ and $\mathrm{CD}=3.5 \mathrm{cm}$.

Write the answer as Yes / No.

In the following figure, $\mathrm{DE}\parallel \mathrm{AC}$ and $\mathrm{DC}\parallel$$\mathrm{AP}.$ prove that: $\frac{\mathrm{BE}}{\mathrm{EC}}=\frac{\mathrm{BC}}{\mathrm{CP}}.$

In $△\mathrm{ABC},\mathrm{D}$ and $\mathrm{E}$ are points on the sides $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{DE}\parallel \mathrm{BC}$, $\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{3}{4}$ and $\mathrm{AC}=15 \mathrm{cm}$. If $\mathrm{AE}=k \mathrm{cm}$, then find the value of $k$. (Round-off the answer to two decimal)

In $△\mathrm{ABC}$, $\mathrm{D}$ and $\mathrm{E}$ are points on the sides $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{DE}\parallel \mathrm{BC}$. If $\mathrm{AD}=8x-7$, $\mathrm{DB}=5x-3$, $\mathrm{AE}=4x-3$ and $\mathrm{EC}=3x-1$, then find the value of $x$.

$\mathrm{ABCD}$ is a trapezium in which $\mathrm{AB}\Vert \mathrm{DC}$ and $\mathrm{P}, \mathrm{Q}$ are points on $\mathrm{AD}$ and $\mathrm{BC}$ respectively, such that $\mathrm{PQ}\Vert \mathrm{DC}$. If $\mathrm{PD}=18 \mathrm{cm}, \mathrm{BQ}=35 \mathrm{cm}$, $\mathrm{QC}=15 \mathrm{cm}$ and $\mathrm{AD}=x \mathrm{cm}$, find the value of $x$.

In the figure,$\mathrm{ABC}$ is a triangle in which $\mathrm{AB}=\mathrm{AC}$ . Points$\mathrm{D}$ and $\mathrm{E}$ are points on the side $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{AD}=\mathrm{AE}$. Show that the points $\mathrm{B}, \mathrm{C}, \mathrm{E}$ and $\mathrm{D}$ lie on a same circle.

$\mathrm{O}$ is any point inside a triangle $\mathrm{ABC}$. The bisector of $\angle \mathrm{AOB} ,$$\angle \mathrm{BOC}$ and $\angle \mathrm{COA}$meet the sides$\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CA}$ at point $\mathrm{D}, \mathrm{E}$ and$\mathrm{F}$ respectively. Show that $\mathrm{AD}\times \mathrm{BE}\times \mathrm{CF}=\mathrm{DB}\times \mathrm{EC}\times \mathrm{FA}$.