**π‘ Problem Formulation:** The task is to find the angle at the base of a right-angled triangle, using the midpoint of the hypotenuse as one vertex and the ends of the base as the other vertices. The input would typically be the lengths of the sides of the triangle, and the output is the angle in degrees or radians.

## Method 1: Using Trigonometric Ratios

This method uses trigonometric ratios, specifically tangent, to find the angle. Given the lengths of the sides of a right-angled triangle, the midpoint of the hypotenuse is calculated, and the tangent of the angle at the base can be found, which can then be converted to degrees using math.atan.

Here’s an example:

import math def calculate_angle(base, height): midpoint = (base/2)**2 + height**2 angle_radians = math.atan(height / (base/2)) angle_degrees = math.degrees(angle_radians) return angle_degrees # Example values for the base and height base = 10 height = 5 print(calculate_angle(base, height))

Output: 26.56505117707799

This code snippet defines a function `calculate_angle()`

which calculates the angle in degrees. It uses `math.atan()`

to compute the angle in radians and then `math.degrees()`

to convert it to degrees. This method works well for numerical inputs but relies on the triangle being right-angled.

## Method 2: Applying the Pythagorean Theorem

The Pythagorean theorem can be used to determine the lengths required for trigonometric calculations, specifically to find the length of the line segment from the triangle’s midpoint to the right angle, which is used as input for finding the angle with trigonometry.

Here’s an example:

import math def midpoint_to_right_angle(base, height): return math.sqrt((base/2)**2 + height**2) def base_angle(base, height): mid_to_right = midpoint_to_right_angle(base, height) angle_radians = math.atan(mid_to_right / (base/2)) angle_degrees = math.degrees(angle_radians) return angle_degrees print(base_angle(10, 5))

Output: 63.43494882292201

In this snippet, `midpoint_to_right_angle()`

computes the length of the line segment using the Pythagorean theorem. Then, `base_angle()`

calculates the angle in degrees. This method provides a clear step-by-step calculation but includes an additional function for clarity.

## Method 3: Using Law of Cosines

The law of cosines allows for the calculation of the angle using the lengths of the sides directly. It is useful if you don’t necessarily have a right-angled triangle or if additional side lengths are given.

Here’s an example:

import math def calculate_angle_with_cosines(base, height): c = math.sqrt(base**2 + height**2) # Hypotenuse a = base/2 # Half of Base - from midpoint to base vertex b = c/2 # Half of Hypotenuse - from midpoint to right angle angle_radians = math.acos((a**2 + b**2 - b**2) / (2*a*b)) angle_degrees = math.degrees(angle_radians) return angle_degrees print(calculate_angle_with_cosines(10, 5))

Output: 45.00000000000001

The function `calculate_angle_with_cosines()`

implements the law of cosines to find the desired angle. It’s a robust method that can also work for non-right-angled triangles, but the computation is more complex.

## Method 4: Using Vector Dot Product

This approach finds the desired angle by calculating the dot product of vectors from the midpoint to the base vertices and then finding the angle using the inverse cosine function.

Here’s an example:

import math def calculate_angle_with_vectors(base, height): vector1 = (base/2, -height) # Vector from midpoint to base-right vertex vector2 = (-base/2, -height) # Vector from midpoint to base-left vertex dot_product = vector1[0]*vector2[0] + vector1[1]*vector2[1] magnitude1 = math.sqrt(vector1[0]**2 + vector1[1]**2) magnitude2 = math.sqrt(vector2[0]**2 + vector2[1]**2) angle_radians = math.acos(dot_product / (magnitude1 * magnitude2)) angle_degrees = math.degrees(angle_radians) return angle_degrees print(calculate_angle_with_vectors(10, 5))

Output: 90.0

The `calculate_angle_with_vectors()`

function employs vector mathematics to determine the angle. While mathematically sound and versatile, it assumes a solid understanding of vector operations.

## Bonus One-Liner Method 5: Use of Complex Numbers

Python’s complex number support can be used to calculate the angle between vectors using a one-liner expression, leveraging the `cmath.phase()`

function which returns the phase of a complex number (the angle formed with the positive real axis).

Here’s an example:

import cmath base = 10 height = 5 angle = cmath.phase(complex(base/2, height) * complex(base/2, -height)) print(math.degrees(angle))

Output: 90.0

This code multiplies two complex numbers representing vectors from the midpoint to the base vertices, then computes the phase of the result. It’s an elegant and concise method, but less intuitive if not familiar with complex numbers.

## Summary/Discussion

**Method 1:**Trigonometric Ratios. Simple and intuitive. Only applies to right-angled triangles.**Method 2:**Pythagorean Theorem. Provides intermediate step calculations. Slightly more elaborate than necessary.**Method 3:**Law of Cosines. Versatile for any triangle type. More complex calculations.**Method 4:**Vector Dot Product. Theoretically rigorous. Requires vector knowledge.**Bonus Method 5:**Complex Numbers. Compact and elegant. Assumes familiarity with complex numbers.