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In triangle ABC, altitude BE = altitude CF. Then triangle ABC is
- a)Equilateral triangle
- b)Scalene
- c)Isosceles triangle
- d)Right angled triangle
Correct answer is option 'C'. Can you explain this answer?
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In triangle ABC, altitude BE = altitude CF. Then triangle ABC isa)Equi...
Understanding the Triangle Properties
In triangle ABC, if the altitudes BE and CF from vertices B and C to the opposite sides are equal, it indicates a special relationship between the sides and angles of the triangle.
Key Observations
- Equal Altitudes: The condition BE = CF means that the perpendicular distances from points B and C to the opposite sides (AC and AB, respectively) are the same.
- Implications on Side Lengths: When altitudes from two different vertices to the opposite sides are equal, this suggests that the lengths of the sides opposite these vertices (AC and AB) must also be equal for the triangle to maintain its shape.
Conclusion: Isosceles Triangle
- Isosceles Triangle Definition: A triangle is classified as isosceles if it has at least two sides that are equal. Since the altitudes BE and CF are equal, the sides AC and AB must be equal, confirming that triangle ABC is indeed isosceles.
- Why Not Other Types?:
- Equilateral Triangle: While all sides are equal, the condition alone does not guarantee this.
- Scalene Triangle: This type has all sides different, contradicting our equal altitude condition.
- Right-Angled Triangle: It can be isosceles, but not necessarily based solely on the equal altitudes.
In summary, the equality of altitudes BE and CF directly leads to the conclusion that triangle ABC is an isosceles triangle, as it satisfies the condition of having at least two equal sides.
In triangle ABC, if the altitudes BE and CF from vertices B and C to the opposite sides are equal, it indicates a special relationship between the sides and angles of the triangle.
Key Observations
- Equal Altitudes: The condition BE = CF means that the perpendicular distances from points B and C to the opposite sides (AC and AB, respectively) are the same.
- Implications on Side Lengths: When altitudes from two different vertices to the opposite sides are equal, this suggests that the lengths of the sides opposite these vertices (AC and AB) must also be equal for the triangle to maintain its shape.
Conclusion: Isosceles Triangle
- Isosceles Triangle Definition: A triangle is classified as isosceles if it has at least two sides that are equal. Since the altitudes BE and CF are equal, the sides AC and AB must be equal, confirming that triangle ABC is indeed isosceles.
- Why Not Other Types?:
- Equilateral Triangle: While all sides are equal, the condition alone does not guarantee this.
- Scalene Triangle: This type has all sides different, contradicting our equal altitude condition.
- Right-Angled Triangle: It can be isosceles, but not necessarily based solely on the equal altitudes.
In summary, the equality of altitudes BE and CF directly leads to the conclusion that triangle ABC is an isosceles triangle, as it satisfies the condition of having at least two equal sides.
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In triangle ABC, altitude BE = altitude CF. Then triangle ABC isa)Equi...
Correct answer is option C
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