Equilateral triangles are drawn on the sides of a right triangle, then the area of the triangle on the hypotenuse is equal to ________ of the areas of the triangles on the other two sides.

Options

A. Product
B. Sum
C. Difference
D. None of these

Answer

To solve the problem, we need to understand the relationship between the areas of the equilateral triangles drawn on the sides of a right triangle. Let's break down the solution step by step.

### Step 1: Understand the Right Triangle
Consider a right triangle with sides \( a \), \( b \) (the two legs), and \( c \) (the hypotenuse).

### Step 2: Draw Equilateral Triangles
Draw equilateral triangles on each side of the right triangle:
- An equilateral triangle on side \( a \)
- An equilateral triangle on side \( b \)
- An equilateral triangle on side \( c \)

### Step 3: Calculate the Area of Equilateral Triangles
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]

Using this formula, we can calculate the areas of the equilateral triangles:
- Area of triangle on side \( a \):
\[ A_a = \frac{\sqrt{3}}{4} a^2 \]
- Area of triangle on side \( b \):
\[ A_b = \frac{\sqrt{3}}{4} b^2 \]
- Area of triangle on hypotenuse \( c \):
\[ A_c = \frac{\sqrt{3}}{4} c^2 \]

### Step 4: Use the Pythagorean Theorem
Since we are dealing with a right triangle, we can use the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]

### Step 5: Relate the Areas
Now, we can relate the area of the triangle on the hypotenuse to the areas of the triangles on the other two sides:
\[ A_c = \frac{\sqrt{3}}{4} c^2 = \frac{\sqrt{3}}{4} (a^2 + b^2) \]
This can be rewritten as:
\[ A_c = \frac{\sqrt{3}}{4} a^2 + \frac{\sqrt{3}}{4} b^2 = A_a + A_b \]

### Step 6: Conclusion
From the above relation, we can conclude that the area of the triangle on the hypotenuse is equal to the sum of the areas of the triangles on the other two sides. Therefore, the answer to the question is:
\[ \text{The area of the triangle on the hypotenuse is equal to the sum of the areas of the triangles on the other two sides.} \]

### Final Answer
The area of the triangle on the hypotenuse is equal to **the sum** of the areas of the triangles on the other two sides.

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