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

Answer

Verified

362.4k+ 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 $\dfrac{{{\text{AE}}}}{{{\text{AC}}}} = \dfrac{{{\text{AE}}}}{{{\text{AE}} + {\text{CE}}}} = \dfrac{{\text{6}}}{{{\text{6}} + {\text{9}}}} = \dfrac{6}{{15}} = \dfrac{2}{5} = \dfrac{{{\text{AD}}}}{{{\text{AB}}}}$.

Last updated date: 28th Sep 2023

â€¢

Total views: 362.4k

â€¢

Views today: 4.62k

Recently Updated Pages

What do you mean by public facilities

Slogan on Noise Pollution

Paragraph on Friendship

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

What is the Full Form of ILO, UNICEF and UNESCO

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers