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Q2:
In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$ (see Fig. 7.30). Show that $\triangle ABC$ is an isosceles triangle in which $AB = AC$.
In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$ (see Fig. 7.30). Show that $\triangle ABC$ is an isosceles triangle in which $AB = AC$.
Solution :
Step 1: Identification of Given Conditions and Goal
Given: In $\triangle ABC$, $AD \perp BC$ and $BD = DC$.
To Prove: $AB = AC$, which implies $\triangle ABC$ is an isosceles triangle.
Step 2: Establishing Congruence in $\triangle ABD$ and $\triangle ACD$
We consider two triangles: $\triangle ABD$ and $\triangle ACD$. We analyze them based on the provided information:
- $BD = DC$ [Given: $AD$ is the bisector of $BC$]
- $\angle ADB = \angle ADC = 90^{\circ}$ [Given: $AD \perp BC$]
- $AD = AD$ [Common side to both triangles]
Step 3: Application of Congruence Criterion
By the SAS (Side-Angle-Side) Congruence Criterion, which states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent:
$\triangle ABD \cong \triangle ACD$ [By SAS Criterion]
Step 4: Deduction of Equality of Corresponding Parts
Since the triangles are congruent, their corresponding parts are equal (CPCT - Corresponding Parts of Congruent Triangles):
$AB = AC$ [By CPCT]
Step 5: Theoretical Conclusion
Because we have demonstrated that $AB = AC$, the definition of an isosceles triangle (a triangle with at least two equal sides) is satisfied for $\triangle ABC$.
Final Solution: Since $\triangle ABD \cong \triangle ACD$, it follows that $AB = AC$ by CPCT, thereby proving that $\triangle ABC$ is an isosceles triangle.
More Questions from Class 9 Mathematics Triangles EXERCISE 7.2
- Q1(i): In an isosceles triangle ABC, with $AB = AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that : (i) $OB = OC$
- Q1(ii): In an isosceles triangle ABC, with $AB = AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that : (ii) $AO$ bisects $\angle A$
- Q3: ABC is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively (see Fig. 7.31). Show that these altitudes are equal.
- Q4(i): ABC is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal (see Fig. 7.32). Show that (i) $\triangle ABE \cong \triangle ACF$
- Q4(ii): ABC is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal (see Fig. 7.32). Show that (ii) $AB = AC$, i.e., ABC is an isosceles triangle.
- Q5: ABC and DBC are two isosceles triangles on the same base $BC$ (see Fig. 7.33). Show that $\angle ABD = \angle ACD$.
- Q6: $\triangle ABC$ is an isosceles triangle in which $AB = AC$. Side $BA$ is produced to $D$ such that $AD = AB$ (see Fig. 7.34). Show that $\angle BCD$ is a right angle.
- Q7: ABC is a right angled triangle in which $\angle A = 90^{\circ}$ and $AB = AC$. Find $\angle B$ and $\angle C$.
- Q8: Show that the angles of an equilateral triangle are $60^{\circ}$ each.
CBSE Solutions for Class 9 Mathematics Triangles
Chapters in CBSE - Class 9 Mathematics
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