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If two isosceles triangles have a common base, then the line joining their vertices will
  • a)
    Bisect them at acute angle
  • b)
    Bisect them at obtuse angle
  • c)
    Bisect them at right angle
  • d)
    NOT
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If two isosceles triangles have a common base, then the line joining t...

Given: ΔABC and ΔBCP are isosceles ΔS with common base BC.
In ΔS ABP and ACP
AB = AC (given)
BP = PC (given)
AP = AP (common)
∴ ΔABP ≅ ΔACP (S-S-S criterion)
∴ BO = OC and
∠AOB = ∠AOC = 180°
⇒ 2∠AOC = 180°
⇒ ∠AOB = 90°
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Community Answer
If two isosceles triangles have a common base, then the line joining t...
Explanation:

To understand why the line joining the vertices of two isosceles triangles with a common base bisects them at a right angle, let's consider the properties of isosceles triangles.

Properties of Isosceles Triangles:
1. In an isosceles triangle, the two sides opposite the equal angles are of equal length.
2. The angles opposite the equal sides of an isosceles triangle are equal.

Given:
We have two isosceles triangles with a common base.

To Prove:
The line joining their vertices bisects the triangles at a right angle.

Proof:
Let's label the triangles as ABC and ABD, where AB is the common base and AC = BC and AD = BD.

Now, let's draw the line joining the vertices C and D.

Step 1:
Since AC = BC and AD = BD, we can conclude that triangles ABC and ABD are congruent by the Side-Side-Side (SSS) congruence criterion.

Step 2:
Using the congruence of the triangles, we can say that angle BAC is congruent to angle BAD and angle ABC is congruent to angle ABD.

Step 3:
Since angles BAC and BAD are congruent, and angles ABC and ABD are congruent, we can conclude that angle CAB is congruent to angle DAB.

Step 4:
When two angles are congruent, they are called vertically opposite angles. Vertically opposite angles are always equal.

Step 5:
Since angle CAB and angle DAB are vertically opposite angles and congruent, we can conclude that the line joining the vertices C and D bisects the triangles ABC and ABD at a right angle.

Hence, the correct answer is option 'C' - The line joining the vertices of two isosceles triangles with a common base bisects them at a right angle.
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