Theorems on Similarity

$ABC$ is a right angled triangle at $A= 90°$, if $AD$ is perpendicular to $BC$, prove that $\frac{∆ABC}{∆ACD}= \frac{B{C}^2}{A{C}^2}$

$ABC$ is a right-angled triangle at $A=90°$. If $AD$ is perpendicular to $BC$, prove that $\frac{\mathrm{ar}\left(∆ABC\right)}{\mathrm{ar}\left(∆ACD\right)}=\frac{B{C}^2}{A{C}^2}$

In the triangle ABC, AD and BE are two medians and DF parallel to BE, meets AC at the point F. If the length of the side AC is $8 cm$., then let us write the length of the side CF.

In the triangle $ABC$, $AD$ and $BE$ are two medians and $DF$ parallel to $BE$ meets $AC$ at the point $F$. If the length of the side $AC$ is $8 \mathrm{cm}$, then find the length of $CF$

In $\Delta ABC$, points $P$ and $Q$ are on$CA$ and $CB$, respectively such that $CA=16 \mathrm{cm}, CP=10 \mathrm{cm},$ $CB=30 \mathrm{cm}$ and $CQ=25 \mathrm{cm}$. Is $PQ\Vert AB$? (Answer in Yes or No)

In the given figure, $ADF$ and $CEF$ are two similar triangles, $BA\parallel CE, DE\parallel BC$ and  $AF: AC= 5: 8$. If $CE=6 \mathrm{cm}$ and $AB=k \mathrm{cm}$, then find the value of $k$.

In the given figure, $\angle \mathrm{ADE}=\angle \mathrm{B}$. If $\mathrm{AD}=3.8 \mathrm{cm}, \mathrm{AE}=2.1 \mathrm{cm}, \mathrm{BE}=3.6 \mathrm{cm}, \mathrm{BC}=4.2 \mathrm{cm}$. If $\mathrm{DE}=k \mathrm{cm}$, then find the value of $k$ (in decimal form).

A model of a ship is made to a scale of $1 : 200$. When the length of the model is $4 \mathrm{m}$, the length of the ship is $k \mathrm{metres}$. Find te value of $k$.

The scale of a map is $1 : 200000$. If the $A {\mathrm{km}}^2$ can be represented by $1 {\mathrm{cm}}^2$ on the map, then find the value of $k$.

In $△ABC, D$ and $E$ are two points on $AB$ and $AC$ and $DE\parallel BC$. If $AD=x, DB=x+2, AE=x-2, EC=x-1,$ then what is the value of $x$?

In an isosceles $△ABC$, if $AB=AC=25 \mathrm{cm}$. If the length of the altitude from $A$ on $BC$ is $24 \mathrm{cm}$ and $BC=k \mathrm{cm}$, then find the value of $k$.

In trapezium $ABCD$, $AB$ and $DC$ are parallel. A straight line parallel to $AB$ meets $AD$ and $BC$ at $E$ and $F$. Then,

$AB$ is a diameter of a circle. The straight line through $A$ meets the circle at $D$ and the tangent at $B$ at $C$. Then,

In triangles $ABC$ and $DEF$, if $\angle A=\angle E=90°$ ; $AB : ED=AC : EF$ and $\angle F=35°$, then the value of $\angle B$ is

A model of a square $PQRS$ is made to a scale of $0.006$. The area of the square on the model is $150 {\mathrm{cm}}^2$. If the actual area of the square is $k {\mathrm{cm}}^2$, then find the value of $k$.

A model of a ship is made to a scale of $1:200$. The volume of the model is $100 \mathrm{L}$. If the volume of the ship is $x \mathrm{L}$, then find the value of $x$. 

$AD$ is a diameter of the circumcircle of $△ABC$. $AE$ is perpendicular to $BC$. Prove that $AB.AC=AD.AE$

A model of a square $PQRS$ is made to a scale of $0.006$. The area of the square on the model is $150 {\mathrm{cm}}^2$. If the actual area of the square is $k {\mathrm{cm}}^2$, then find the value of $k$.

A model of a ship is made to a scale of $1:200$. The volume of the model is $100 \mathrm{L}$. If the volume of the ship is $x \mathrm{L}$, then find the value of $x$. 

In $∆DEW, AB \left|\right| EW$. If $AD = 4 \mathrm{cm}$, $DE = 12 \mathrm{cm}$ and $DW = 24 \mathrm{cm}$, then find the value of $DB$.