If the medians of a triangle are equal, then the triangle will be:

Correct — D. Rule: a median's length is fixed by the sides — 4m² = 2b² + 2c² − a² for the median m drawn to side a (Apollonius).

Write it for two medians and set them equal.

Median to a: 4m² = 2b² + 2c² − a²
Median to b: 4m² = 2a² + 2c² − b²

Equal ⇒ 2b² − a² = 2a² − b² ⇒ 3b² = 3a² ⇒ a = b

The same step on the other pair gives b = c, so all three sides match and the triangle is equilateral → option (d)

Why the others are wrong

• (a)
  right angled triangle — A right angle puts no constraint on the medians. In a 3–4–5 triangle they measure about 4.27, 3.61 and 2.5 units — three different values — so right-angled does not follow from equal medians.
• (b)
  obtuse angle triangle — An obtuse triangle is ruled out for the same reason as the others: equal medians force equal sides, and equal sides force three 60° angles. No angle above 90° can survive that.
• (c)
  scalene triangle — Scalene means all three sides different — the exact opposite of what equal medians produce. It tempts only if you read the condition as the medians being unequal.

A median joins a vertex to the midpoint of the opposite side, and Apollonius' theorem ties its length to the three sides: 4m² = 2b² + 2c² − a².

That formula is symmetric in b and c, so two medians come out equal exactly when the two sides they are drawn to are equal. Equal medians and equal sides are one condition in two languages.

Two equal medians therefore give an isosceles triangle. The question makes all three equal, and that is what forces equilateral.

Key facts

• A median joins a vertex to the midpoint of the opposite side, and the three medians meet at the centroid.
• Apollonius: 4m² = 2b² + 2c² − a², where m is the median drawn to side a.
• Two equal medians make a triangle isosceles, and three equal medians make it equilateral.
• The centroid divides every median in the ratio 2 : 1, measured from the vertex.

Common traps

• Answering scalene by reading the condition backwards
• Assuming medians are a right-triangle topic because of the hypotenuse formula
• Stopping at isosceles, which two equal medians would give, when the stem makes all three equal

No directly related past PYQ was found.