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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) $OB = OC$
Solution :
Step 1: Identifying Given Parameters and Properties
Given an isosceles triangle $\triangle ABC$ where $AB = AC$. According to the Isosceles Triangle Theorem, the angles opposite to the equal sides are equal. Therefore:
$\angle ABC = \angle ACB$ [Angles opposite to equal sides of a triangle are equal]
Step 2: Utilizing Angle Bisector Definitions
We are given that $OB$ is the bisector of $\angle ABC$ and $OC$ is the bisector of $\angle ACB$. By the definition of an angle bisector:
- $\angle OBC = \frac{1}{2} \angle ABC$
- $\angle OCB = \frac{1}{2} \angle ACB$
Since $\angle ABC = \angle ACB$ (from Step 1), it follows by substituting these relations that:
$\angle OBC = \angle OCB$
Step 3: Applying the Converse of the Isosceles Triangle Theorem
Consider $\triangle OBC$. We have established that $\angle OBC = \angle OCB$.
According to the Converse of the Isosceles Triangle Theorem, if two angles of a triangle are equal, then the sides opposite to those angles are also equal. Therefore:
$OB = OC$ [Sides opposite to equal angles of a triangle are equal]
Visual Representation of Triangle OBC
Summary of Logic
| Premise | Deduction |
|---|---|
| $AB = AC$ | $\angle ABC = \angle ACB$ |
| $OB, OC$ are bisectors | $\angle OBC = \angle OCB$ |
| $\angle OBC = \angle OCB$ | $OB = OC$ |
Final Solution: Since $\angle OBC = \angle OCB$, the sides opposite to these angles in $\triangle OBC$ must be equal; hence, $OB = OC$.
More Questions from Class 9 Mathematics Triangles EXERCISE 7.2
- 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 $\angle A$
- 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 $AB = AC$.
- 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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