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In the figure given below, ABC is a triangle with AB = BC and D is an interior point of the triangle ABC such that ∠DAC = ∠DCA.
Consider the following statements:
1. Triangle ADC is an isosceles triangle.
2. D is the centroid of the triangle ABC.
3. Triangle ABD is congruent to the triangle CBD.
Q. Which of the above statements are correct?
  • a)
    1 and 2 only
  • b)
    2 and 3 only
  • c)
    1 and 3 only
  • d)
    1, 2 and 3
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In the figure given below, ABC is a triangle with AB = BC and D is an ...

 
Statement 1: Triangle ADC is an isosceles triangle.
Given ∠DAC = ∠DCA;
∴ DA = DC
Since two sides of the triangle are equal, triangle ADC is an isosceles triangle.
∴ Statement 1 is correct.
Statement 2: D is the centroid of the triangle ABC.
Point D can be anywhere in the line BE (It will satisfy the given condition too i.e. DA = DC)
∴ D may or may not be the centriod of the triangle ABC.
∴ Statement 2 is not correct.
Statement 3: Triangle ABD is congruent to the triangle CBD.
Since AD = CD and AB = CB;
∴ Triangle ABD is congruent to the triangle CBD
∴ Statement 3 is correct.
∴ Statement 1 and 3 are correct.
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In the figure given below, ABC is a triangle with AB = BC and D is an interior point of the triangle ABC such that ∠DAC = ∠DCA.Consider the following statements:1. Triangle ADC is an isosceles triangle.2. D is the centroid of the triangle ABC.3. Triangle ABD is congruent to the triangle CBD.Q. Which of the above statements are correct?a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'C'. Can you explain this answer?
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