   Question Answers

# In a triangle ABC, DE is parallel to BC. If $\dfrac{{AD}}{{DB}} = \dfrac{2}{3}$ and AC = 18 cm, then find AE.  Hint: Draw triangle ABC and DE parallel to BC. Prove that the two triangles are similar and then use the properties of similar triangles to find the relation between AE and AC and then find AE. In the diagram, ABC is a triangle and the line segment DE is parallel to BC.
Let us check triangle ABC and triangle ADE for similarity.
We know that corresponding angles on two parallel lines are equal. Hence, angle ADE is equal to angle ABC and angle AED is equal to angle ACB.
$\angle ADE = \angle ABC..............(1)$
$\angle AED = \angle ACB..............(2)$
Angle A is common between both the triangles. Hence, we have:
$\angle EAD = \angle BAC..............(3)$
From equation (1), equation (2) and equation (3), the triangles ABC and ADE are similar by Angle-Angle-Angle (AAA) criterion. Hence, we have:
$\Delta ABC \sim \Delta ADE$
We know that the properties of similar triangles states that the ratio of corresponding sides of similar triangles are equal.
For triangle ABC and triangle ADE, we have AB and AD, AC and AE, BC and DE as corresponding sides. Hence, the ratio of AB by AD is equal to the ratio of AC by AE is equal to the ratio of BC by DE.
Hence, we have as follows:
$\dfrac{{AB}}{{AD}} = \dfrac{{AC}}{{AE}} = \dfrac{{BC}}{{DE}}$
Now, considering the first two equal terms we have:
$\dfrac{{AB}}{{AD}} = \dfrac{{AC}}{{AE}}$
We know that AB = AD + DB, then, we have:
$\dfrac{{AD + DB}}{{AD}} = \dfrac{{AC}}{{AE}}$
Simplifying, we have:
$1 + \dfrac{{DB}}{{AD}} = \dfrac{{AC}}{{AE}}$
It is given that $\dfrac{{AD}}{{DB}} = \dfrac{2}{3}$ and AC = 18 cm. Hence, we have:
$1 + \dfrac{3}{2} = \dfrac{{18}}{{AE}}$
Simplifying, we have:
$\dfrac{5}{2} = \dfrac{{18}}{{AE}}$
Solving for AE, we have:
$AE = \dfrac{2}{5} \times 18$
$AE = 7.2cm$
Hence, the value of AE is 7.2 cm.

Note: You can also prove the similarity by other criteria. The ratio of AD to DB is only given, hence, you can either express AB as AD + DB or you can solve to find the ratio of AB to AD.
View Notes
De Morgan's First Law  Capacitors in Parallel  Where There is a Will There is a Way Essay  Resistors in Series and Parallel  CBSE Class 10 Maths Chapter 6 - Triangles Formula  Wave Nature of Matter and De Broglie’s Equation  CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry Formula  Indian Olympiad Qualifier in Mathematics- IOQM (Similar to PRMO of Last Year)  CBSE Class 10 Maths Chapter 13 - Surface Areas and Volumes Formula  The Difference Between an Animal that is A Regulator and One that is A Conformer  Important Questions for CBSE Class 10 Maths Chapter 12 - Areas Related to Circles  Important Questions for CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry  Important Questions for CBSE Class 10 Maths Chapter 6 - Triangles  Important Questions for CBSE Class 8 English Honeydew Chapter 7 - A Visit to Cambridge  Important Questions for CBSE Class 9 Maths Chapter 9 - Areas of Parallelograms and Triangles  Important Questions for CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables  Important Questions for CBSE Class 10 Maths Chapter 13 - Surface Areas and Volumes  Important Questions for CBSE Class 10 Maths Chapter 10 - Circles  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 10 - A Strange Wrestling Match  Important Questions for CBSE Class 11 Physics Chapter 4 - Motion in a Plane  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  Hindi A Class 10 CBSE Question Paper 2016  Hindi A Class 10 CBSE Question Paper 2012  Hindi A Class 10 CBSE Question Paper 2010  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  Hindi A Class 10 CBSE Question Paper 2008  Hindi A Class 10 CBSE Question Paper 2014  RS Aggarwal Class 9 Solutions Chapter-9 Congruence of Triangles and Inequalities in a Triangle  NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles  RD Sharma Solutions for Class 10 Maths Chapter 15 - Areas Related to Circles  NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles in Hindi  NCERT Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles  RD Sharma Solutions for Class 9 Maths Chapter 15 - Areas of Parallelograms and Triangles  NCERT Solutions for Class 10 Maths Chapter 6 Triangles  NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry in Hindi  RS Aggarwal Class 10 Solutions - Triangles  