ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE.
Solution:
Given, ABC is an isosceles triangle.
AB = AC
BD and CE are the medians
We have to show that BD = CE
CE is the median implies that E is the midpoint of AB
AB = AE + BE
So, AB = 2AE ----------------- (1)
BD is the median implies that D is the midpoint of AC
AC = AD + CD
So, AC = 2AD ----------------- (2)
Consider triangles ABD and ACE,
AB = AC (given)
∠A = common angle
AB = AC
From (1) and (2),
2AE = 2AD
AE = AD
SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are congruent.
By SAS criterion, ΔABD ≅ ΔACE
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.
Considering triangles ABD and ACE,
By CPCT,
BD = CE
Therefore, it is proven that BD = CE
✦ Try This: For ΔABC and ΔDEF, AB=FE,BC=ED and ∠B=∠E. Are the triangles similar?
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.3 Problem 1
ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE
Summary:
An isosceles triangle is a type of triangle that has any two sides equal in length. ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. It is shown that BD = CE
☛ Related Questions:
- In Fig.7.4, D and E are points on side BC of a ∆ ABC such that BD = CE and AD = AE. Show that ∆ ABD . . . .
- CDE is an equilateral triangle formed on a side CD of a square ABCD (Fig.7.5). Show that ∆ ADE ≅ ∆ B . . . .
- In Fig.7.6, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that ∆ ABC ≅ ∆ DEF