If the medians of a triangle are equal, then the triangle is:a)right angledb)isoscelesc)equilaterald)scaleneCorrect answer is 'C'. Can you explain this answer?

Question

Answer

In equilateral triangle median this can be prove by Congruency condition

Answer

Median means from a vertices to the mid point of the opposite side of the triangle. if it the 3 medians of a triangle give the same answer. it means the side are also of the same length. and if the sides are same it is known as an equilateral triangle

Answer

Explanation:

The medians of a triangle are the line segments that connect each vertex of the triangle to the midpoint of the opposite side. Each triangle has three medians, and they intersect at a point called the centroid. The centroid divides each median into two segments, with the segment connecting the centroid to the vertex being twice as long as the segment connecting the centroid to the midpoint of the opposite side.

Definition of an equilateral triangle:
An equilateral triangle is a triangle in which all three sides are equal in length. In an equilateral triangle, all three angles are also equal, measuring 60 degrees.

Proof:
To prove that if the medians of a triangle are equal, then the triangle is equilateral, we can use the property of the centroid. Since the medians of a triangle intersect at the centroid, and the segment connecting the centroid to the vertex is twice as long as the segment connecting the centroid to the midpoint of the opposite side, if the medians are equal, then all these segments are equal in length.

Step 1: Assume that the medians of the triangle are equal.
Let's assume that the medians of the triangle are equal and denote their length as 'm'.

Step 2: Use the property of the centroid.
According to the property of the centroid, the segment connecting the centroid to the vertex is twice as long as the segment connecting the centroid to the midpoint of the opposite side. Therefore, the length of the segment connecting the centroid to the vertex is 2m.

Step 3: Use the definition of an equilateral triangle.
In an equilateral triangle, all three sides are equal in length. Therefore, if the length of the segment connecting the centroid to the vertex is 2m, then all three sides of the triangle are 2m. Hence, the triangle is equilateral.

Conclusion:
In conclusion, if the medians of a triangle are equal, then the triangle is equilateral. This is because the equality of the medians implies the equality of all three sides of the triangle, which is the defining property of an equilateral triangle.

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