MCQ
$\triangle\text{ABC}$ and $\triangle\text{BDE}$ are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangle ABC and BDE is:
- A2 : 1
- B1 : 2
- ✓4 : 1
- D1 : 4
✓
Answer
Correct option: C.
4 : 1
$\triangle\text{ABC}$ and $\triangle\text{BDE}$ are equilateral triangles and D is the mid-point of PC.
$\triangle\text{ABC}$ and $\triangle\text{BDE}$ are both equilateral triangles
$\therefore$ They are similar also
$\therefore\frac{\text{area of }\triangle\text{ABC}}{\text{area of }\triangle\text{BDE}}=\frac{\text{BC}^2}{\text{BD}^2}=\frac{\text{BC}^2}{\big(\frac{1}{2}\text{BC}^2\big)}$ {D is mid point of BC}
$=\frac{\text{BC}^2}{\frac{1}{4}\text{BC}^2}=\frac{\text{4BC}^2}{\text{BC}^2}=\frac{4}{1}$
$\therefore$ Ratio is 4 : 1
$\triangle\text{ABC}$ and $\triangle\text{BDE}$ are both equilateral triangles
$\therefore$ They are similar also
$\therefore\frac{\text{area of }\triangle\text{ABC}}{\text{area of }\triangle\text{BDE}}=\frac{\text{BC}^2}{\text{BD}^2}=\frac{\text{BC}^2}{\big(\frac{1}{2}\text{BC}^2\big)}$ {D is mid point of BC}
$=\frac{\text{BC}^2}{\frac{1}{4}\text{BC}^2}=\frac{\text{4BC}^2}{\text{BC}^2}=\frac{4}{1}$
$\therefore$ Ratio is 4 : 1
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the given figure, a circle touches the side $B C$ of $\triangle A B C$ at $P$ and touches $A B$ and $A C$ produced at $Q$ and $R$ respectively. If $A Q=5 \ cm$, then find the perimeter of $\triangle A B C$.
→$\sum(\text{x}_\text{i}-\bar{\text{x}})$ is equal to :
→In the given figure, if $P Q \| B C$. Find $A Q$.
→If $a \cot \theta+b \operatorname{cosec} \theta=p$ and $b \cot \theta+a \operatorname{cosec} \theta=q$, then $p^2-q^2=$
→In the given figure, O is the centre of a circle; PQL and PRM are the tangents at the points Q and R respectively, and S is a point on the circle, such that $\angle\text{SQL} = 50^\circ.$ and $\angle\text{SRM} = 60^\circ.$ Find $ \angle\text{QSR}=?$
→The common difference of the $A.P$ whose $S_n= 3n_2+ 7n$ is :
→If one zero of the quadratic polynomial $(k - 1) x^2+ kx + 1$ is $-4,$ then the value of $k$ is :
→Euclid's division lemma states that for two positive integers $a$ and $b$, there exist unique integers $q$ and r such that $a = bq + r$, where r must satisfy
→In the figure, a circle touches the side DF of $\triangle\text{EDF}$ at H and touches ED and EF produced at K and M respectively. If EK = 9cm, then the perimeter $\triangle\text{EDF}$ of is:
→A die is thrown once. Find the probability of getting a number between $3$ and $6$.
→