D, E and F are respectively the mid-points of sides AB, BC and CA of $∆ABC$. Find the ratio of the areas of $∆DEF$ and $∆ABC$.
Given:
D, E and F are respectively the mid-points of sides AB, BC and CA of $∆ABC$.
To do:
We have to find the ratio of the areas of $∆DEF$ and $∆ABC$.
Solution:
$D$ and $E$ are the mid-points of sides $AB$ and $BC$ respectively.
We know that,
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
This implies,
$EF=\frac{1}{2}AB$
$DF=\frac{1}{2}BC$
$DE=\frac{1}{2}AC$
Therefore,
$\frac{\mathrm{DE}}{\mathrm{AC}}=\frac{\mathrm{EF}}{\mathrm{AB}}=\frac{\mathrm{DF}}{\mathrm{BC}}=\frac{1}{2}$
Therefore, by SSS criterion,
$\Delta \mathrm{DEF} \sim \triangle \mathrm{ABC}$
This implies,
$\frac{\operatorname{ar} \triangle \mathrm{DEF}}{\text { ar } \triangle \mathrm{ABC}}=\frac{\mathrm{DE}^{2}}{\mathrm{AC}^{2}}$
$=(\frac{1}{2})^{2}$
$=\frac{1}{4}$
The ratio of the areas of $∆DEF$ and $∆ABC$ is $1: 4$.
Related Articles
- ∆ABC ~ ∆LMN. In ∆ABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ∆ABC and ∆LMN such that $\frac{BC}{MN} \ =\ \frac{5}{4}$.
- In a $\triangle ABC, D, E$ and $F$ are respectively, the mid-points of $BC, CA$ and $AB$. If the lengths of sides $AB, BC$ and $CA$ are $7\ cm, 8\ cm$ and $9\ cm$, respectively, find the perimeter of $\triangle DEF$.
- Let $∆ABC \sim ∆DEF$ and their areas be respectively $64\ cm^2$ and $121\ cm^2$. If $EF = 15.4\ cm$, find BC.
- In $\triangle ABC$, the coordinates of vertex $A$ are $(0, -1)$ and $D (1, 0)$ and $E (0, 1)$ respectively the mid-points of the sides $AB$ and $AC$. If $F$ is the mid-point of side $BC$, find the area of $\triangle DEF$.
- If $∆ABC ≅ ∆FED$ under the correspondence $ABC\Leftrightarrow FED$, write all the corresponding congruent parts of the triangles.
- CD and GH are respectively the bisectors of $\angle ACB$ and $\angle EGF$ such that D and H lie on sides AB and FE of $∆ABC$ and $∆EFG$ respectively. If $∆ABC \sim ∆FEG$, show that (i) \( \frac{\mathrm{CD}}{\mathrm{GH}}=\frac{\mathrm{AC}}{\mathrm{FG}} \)>(ii) \( \triangle \mathrm{DCB} \sim \triangle \mathrm{HGE} \)>(iii) \( \triangle \mathrm{DCA} \sim \triangle \mathrm{HGF} \)
- The vertices of a $\triangle ABC$ are $A(4, 6), B(1, 5)$ and $C(7, 2)$. A line is drawn to intersect sides $AB$ and $AC$ at $D$ and $E$ respectively. such that $\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}$. Calculate the area of the $∆ADE$ and compare it with the area of $∆ABC$.
-  If $D( −51,\ 25 )$, $E( 7,\ 3)$ and $F( 27,\ 27)$ are the mid-points of sides of $\vartriangle ABC$, find the area of $\vartriangle ABC$.
- If $∆DEF ≅ ∆BCA$, write the part(s) of $∆BCA$ that correspond to$(i).\ \angle E$$(ii).\ \overline{EF}$$(iii).\ \angle F$$(iv).\ \overline{DF}$
- Let $ABC$ be an isosceles triangle in which $AB = AC$. If $D, E, F$ are the mid-points of the sides $BC, CA$ and $AB$ respectively, show that the segment $AD$ and $EF$ bisect each other at right angles.
- $S$ and $T$ are points on sides $PR$ and $QR$ of $∆PQR$ such that $\angle P = \angle RTS$. Show that $∆RPQ \sim ∆RTS$.
- If $\triangle ABC$ and $\triangle BDE$ are equilateral triangles, where $D$ is the mid-point of $BC$, find the ratio of areas of $\triangle ABC$ and $\triangle BDE$.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies