Check if right triangle possible from given area and hypotenuse in Python

Given the hypotenuse and area of a right triangle, we need to find the base and height. If it's not possible to form such a triangle, we return False.

So, if the input is like hypo = 10, area = 24, then the output will be (6, 8).

Algorithm

To solve this, we will follow these steps −

• Calculate the maximum possible area for the given hypotenuse
• If the required area exceeds the maximum, return False
• Use binary search to find the base that gives the required area
• Calculate the corresponding height using the Pythagorean theorem

Understanding the Maximum Area

For a right triangle with a fixed hypotenuse, the maximum area occurs when the triangle is isosceles (base = height). In this case, both legs equal hypo / ?2.

Implementation

from math import sqrt

def calculate_area(base, hypotenuse):
    """Calculate area of right triangle given base and hypotenuse"""
    height = sqrt(hypotenuse * hypotenuse - base * base)
    return 0.5 * base * height

def solve(hypo, area):
    """Find base and height of right triangle given hypotenuse and area"""
    hypo_sq = hypo * hypo
    s = sqrt(hypo_sq / 2.0)  # Base when triangle is isosceles
    maxArea = calculate_area(s, hypo)
    
    # Check if required area is possible
    if area > maxArea:
        return False
    
    # Binary search for the correct base
    left = 0.0
    right = s
    
    while abs(right - left) > 0.000001:
        base = (left + right) / 2.0
        current_area = calculate_area(base, hypo)
        
        if current_area >= area:
            right = base
        else:
            left = base
    
    # Calculate final base and height
    height = round(sqrt(hypo_sq - base * base))
    base = round(base)
    return base, height

# Test the function
hypo = 10
area = 24
result = solve(hypo, area)
print(f"Base and Height: {result}")

Base and Height: (6, 8)

Testing Edge Cases

Let's test when the triangle is not possible ?

# Test impossible case
hypo = 5
area = 20  # Too large for hypotenuse of 5

result = solve(hypo, area)
print(f"Result for impossible triangle: {result}")

# Test maximum area case (isosceles triangle)
hypo = 10
max_area = calculate_area(sqrt(50), 10)  # sqrt(100/2) = sqrt(50)
print(f"Maximum area for hypotenuse 10: {max_area:.2f}")

result = solve(hypo, max_area)
print(f"Isosceles triangle result: {result}")

Result for impossible triangle: False
Maximum area for hypotenuse 10: 25.00
Isosceles triangle result: (7, 7)

How It Works

The algorithm uses binary search because the area function is monotonic with respect to the base. As the base increases from 0 to the maximum value, the area first increases and then decreases, reaching maximum when base equals height.

Conclusion

This solution efficiently finds the base and height of a right triangle using binary search. The key insight is that the maximum area occurs when the triangle is isosceles, which helps us determine if the required area is achievable.