# D and E are the points on the sides AB and AC respectively of a triangle ABC such that AD=8 cm, DB=12 cm, AE=6 cm and CE=9 cm. Prove that ${\text{BC}} = \dfrac{{5\left( {{\text{DE}}} \right)}}{2}$ .

Last updated date: 20th Mar 2023

•

Total views: 305.4k

•

Views today: 5.84k

Answer

Verified

305.4k+ views

**Hint:**Here, we will be proceeding by making the two triangles i.e., $\vartriangle {\text{ADE}}$ and $\vartriangle {\text{ABC}}$ as congruent triangles with the help of Inverse of Basic Proportionality Theorem and concept of corresponding angles.

**Complete step-by-step answer:**

Given, AD=8 cm, DB=12 cm, AE=6 cm and CE=9 cm

To prove: ${\text{BC}} = \dfrac{{5\left( {{\text{DE}}} \right)}}{2}$

If we observe the ratio of AD to DB, it is equal to

$ \dfrac{AD}{DB} = \dfrac{8}{{12}} $

$ \Rightarrow \dfrac{AD}{{DB}} = \dfrac{2}{3} $

Now observe the ratio of AE to CE, it is equal to

$ \dfrac{{{\text{AE}}}}{{{\text{CE}}}} = \dfrac{6}{9} $

$ \Rightarrow \dfrac{{{\text{AE}}}}{{{\text{CE}}}} = \dfrac{2}{3}{\text{ }} \to {\text{(2)}} $

As we know that the Inverse of Basic Proportionality Theorem states that if a line divides two sides of a triangle at distinct points in the same ratio then that line is parallel to the third side of the triangle.

By equations (1) and (2), we can say the ratio in which line DE is dividing the two sides AB and AC of the triangle ABC is equal. So, according to the Inverse of Basic Proportionality Theorem this line DE should be parallel to the third side (i.e., BC) of the triangle ABC.

So, DE is parallel to BC i.e., ${\text{DE}}\parallel {\text{BC}}$

Also we know that if ${\text{DE}}\parallel {\text{BC}}$ then the corresponding angles will be equal.

i.e., $\angle {\text{ADE}} = \angle {\text{ABC}}$ and $\angle {\text{AED}} = \angle {\text{ACB}}$

In triangles $\vartriangle {\text{ADE}}$ and$\vartriangle {\text{ABC}}$,

$\angle {\text{ADE}} = \angle {\text{ABC}}$

$\angle {\text{AED}} = \angle {\text{ACB}}$

$\angle {\text{A}} = \angle {\text{A}}$ common to both the triangles $\vartriangle {\text{ADE}}$ and $\vartriangle {\text{ABC}}$

Therefore, by AAA (Angle-Angle-Angle) congruence rule, we can say that both of these triangles are congruent to each other.

i.e., $\vartriangle {\text{ADE}} \cong \vartriangle {\text{ABC}}$

Also we know that, if the two triangles are congruent to each other then, the ratio of their corresponding sides will also be equal.

i.e.,

$ \dfrac{{{\text{AD}}}}{{{\text{AB}}}} = \dfrac{{{\text{DE}}}}{{{\text{BC}}}}$

$ \Rightarrow \dfrac{{{\text{AD}}}}{{{\text{AD + DB}}}} = \dfrac{{{\text{DE}}}}{{{\text{BC}}}} $

$ \Rightarrow \dfrac{{\text{8}}}{{{\text{8 + 12}}}} = \dfrac{{{\text{DE}}}}{{{\text{BC}}}} $

$ \Rightarrow \dfrac{{{\text{DE}}}}{{{\text{BC}}}} = \dfrac{{\text{8}}}{{{\text{20}}}} = \dfrac{{\text{2}}}{{\text{5}}} $

$ \Rightarrow {\text{BC}} = \dfrac{{{\text{5}}\left( {{\text{DE}}} \right)}}{{\text{2}}} $

The above equation represents the required relation between BC and DE.

**Note:**In this particular problem, we have considered the ratios $\dfrac{{{\text{AD}}}}{{{\text{AB}}}}$ and $\dfrac{{{\text{DE}}}}{{{\text{BC}}}}$ to be equal in order to obtain the required relationship between BC and DE but we will be getting same results if we would have considered the ratios $\dfrac{{{\text{AE}}}}{{{\text{AC}}}}$ and $\dfrac{{{\text{DE}}}}{{{\text{BC}}}}$ to be equal because $\dfrac{{{\text{AE}}}}{{{\text{AC}}}} = \dfrac{{{\text{AE}}}}{{{\text{AE}} + {\text{CE}}}} = \dfrac{{\text{6}}}{{{\text{6}} + {\text{9}}}} = \dfrac{6}{{15}} = \dfrac{2}{5} = \dfrac{{{\text{AD}}}}{{{\text{AB}}}}$.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE