Calculating the Area of a Special Trapezoid With Right and 45° Angles

When dealing with geometric shapes and their properties, the trapezoid can sometimes present unique challenges and interests, especially when it comes to calculating its area. In this article, we will explore the process of finding the area of a trapezoid that has a right angle and a 45° angle, where the shorter base is of length a and the longer diagonal is of length 4a. Let's delve into the step-by-step process and understand how to approach such a problem.

Understanding the Trapezoid Configuration

To begin, let's denote and understand the given elements of the trapezoid:

The shorter base b_1 a The longer base b_2, which we will determine The height h, which we will also determine One angle is 45° and another is 90°, visualizing ABCD where AB b_1 a, CD b_2, AD h, and AC (diagonal) 4a

Setting Up the Geometry

Given that triangle ACD is a right triangle with one angle at 45° and an angle at 90°, it forms a 45-45-90 triangle. In such triangles, the lengths of the legs are equal:

h x AD x

Using the Pythagorean theorem, we can find the length of the diagonal:

AC^2 AD^2 CD^2

Relating the Sides

Using the properties of the trapezoid, we can express the length of the longer base b_2 in terms of h:

b_2 a h

Applying the Pythagorean Theorem

Considering the right triangle triangle ACD, we can substitute the known values into the Pythagorean theorem:

4a^2 h^2 (a h)^2

Solving for the Height h

After expanding and rearranging the equation, we get:

2h^2 - 2ah - 15a^2 0

Dividing the entire equation by 2:

h^2 - ah - frac{15}{2}a^2 0

Using the quadratic formula to solve for h (taking the positive solution only):

h frac{a sqrt{31} - 1}{2}

Calculating the Longer Base b_2

Substituting the value of h back into the expression for b_2:

b_2 a frac{a sqrt{31} - 1}{2}

Further simplification gives:

b_2 frac{1 sqrt{31}}{2}a

Calculating the Area of the Trapezoid

The area A of a trapezoid is given by:

A frac{1}{2} b_1 b_2 h

Substituting the known values:

A frac{1}{2} a left( frac{1 sqrt{31}}{2}a right) left( frac{a sqrt{31} - 1}{2} right)

Further simplification results in:

A frac{a^2 (3 sqrt{31})sqrt{31} - 1}{8}

Therefore, the final expression for the area of the trapezoid is: