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In a triangle, if three altitudes are equal, then the triangle is (SSC CGL 1st Sit. 2013)
- a)uilateral
- b)Isoceles
- c)Obtuse
- d)EqRight
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
In a triangle, if three altitudes are equal, then the triangle is (SS...
Let ΔABC is a equilateral triangle of side AB = BC = AC = 2a unit.
AD, BE and CF are three altitudes in the DABC,
In equilateral triangle ∠A = ∠B = ∠C = 60°
Similarly, In ΔACF,
Here, we get AD = BE = CF
Hence, three altitude are equal
Thus, triangle must be a equilateral triangle.
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In a triangle, if three altitudes are equal, then the triangle is (SS...
Understanding the Triangle with Equal Altitudes
When three altitudes in a triangle are equal, the triangle must be equilateral. Here's a detailed explanation of why this is true:
Definition of Altitude
- An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
- In a triangle, the length of the altitude can vary based on the triangle's shape and size.
Implication of Equal Altitudes
- If all three altitudes are equal, it indicates a certain level of symmetry in the triangle.
- This symmetry can only occur in an equilateral triangle, where all sides and angles are equal.
Mathematical Justification
- In an equilateral triangle, since all sides are equal, the altitude from any vertex to the opposite side is the same length.
- For triangles with different side lengths (isosceles or scalene), the altitudes would inherently differ due to the varying base lengths and angles.
Conclusion
- Therefore, the condition of having three equal altitudes leads to the conclusion that the triangle must be equilateral.
- This unique property of equal altitudes is a definitive characteristic of equilateral triangles, ensuring that the correct answer to the question is option 'A': Equilateral.
When three altitudes in a triangle are equal, the triangle must be equilateral. Here's a detailed explanation of why this is true:
Definition of Altitude
- An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
- In a triangle, the length of the altitude can vary based on the triangle's shape and size.
Implication of Equal Altitudes
- If all three altitudes are equal, it indicates a certain level of symmetry in the triangle.
- This symmetry can only occur in an equilateral triangle, where all sides and angles are equal.
Mathematical Justification
- In an equilateral triangle, since all sides are equal, the altitude from any vertex to the opposite side is the same length.
- For triangles with different side lengths (isosceles or scalene), the altitudes would inherently differ due to the varying base lengths and angles.
Conclusion
- Therefore, the condition of having three equal altitudes leads to the conclusion that the triangle must be equilateral.
- This unique property of equal altitudes is a definitive characteristic of equilateral triangles, ensuring that the correct answer to the question is option 'A': Equilateral.
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