Finding r² for Right Angle Triangle Sides in Geometric Progression

We are given a right angle triangle with side lengths \(a, ar, ar^2\), where \(r < 1\). These sides are in a Geometric Progression with the common ratio \(r\). We need to find the Value of r squared.

In any right angle triangle, the longest side is the hypotenuse. Since \(r < 1\), and assuming \(a > 0\), we know that \(r^2 < r < 1\). Therefore, the side lengths in increasing order are \(ar^2, ar, a\). The longest side, or hypotenuse, is \(a\).

This problem involves finding the ratio for Right Angle Triangle Sides.

Applying the Pythagorean Theorem to the Right Angle Triangle Sides

The Pythagorean Theorem states that in a right angle triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Applying the Pythagorean Theorem to this case, for our triangle with sides \(ar^2, ar, a\), this gives us:

$$(ar^2)^2 + (ar)^2 = a^2$$

Let's expand and simplify this equation:

$$a^2 r^4 + a^2 r^2 = a^2$$

Since \(a\) is a side length, \(a \neq 0\). We can safely divide all terms by \(a^2\):

$$\frac{a^2 r^4}{a^2} + \frac{a^2 r^2}{a^2} = \frac{a^2}{a^2}$$

$$r^4 + r^2 = 1$$

Solving the Resulting Quadratic Equation for r²

We can rearrange the equation \(r^4 + r^2 = 1\) into a form that looks like a Quadratic Equation:

$$r^4 + r^2 - 1 = 0$$

This is a Quadratic Equation in terms of \(r^2\). Let \(x = r^2\). Substituting \(x\) into the equation gives us:

$$x^2 + x - 1 = 0$$

We can solve this quadratic equation for \(x\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=1\), and \(c=-1\).

$$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}$$

$$x = \frac{-1 \pm \sqrt{1 + 4}}{2}$$

$$x = \frac{-1 \pm \sqrt{5}}{2}$$

So, the possible values for \(x = r^2\) are \(\frac{-1 + \sqrt{5}}{2}\) and \(\frac{-1 - \sqrt{5}}{2}\).

Determining the Valid Value of r²

Since \(r\) is the ratio of side lengths, \(r\) must be a real number. This means \(r^2\) must be a non-negative value (\(r^2 \ge 0\)).

• The first possible value is \(\frac{-1 + \sqrt{5}}{2}\). Since \(\sqrt{5}\) is approximately 2.236, \(\frac{-1 + \sqrt{5}}{2} \approx \frac{-1 + 2.236}{2} = \frac{1.236}{2} \approx 0.618\). This value is positive, so it is a valid possibility for the Value of r squared.
• The second possible value is \(\frac{-1 - \sqrt{5}}{2}\). Since \(\sqrt{5}\) is approximately 2.236, \(\frac{-1 - \sqrt{5}}{2} \approx \frac{-1 - 2.236}{2} = \frac{-3.236}{2} \approx -1.618\). This value is negative, which is not possible for a real number squared.

Therefore, the only valid Value of r squared is \(\frac{\sqrt{5} - 1}{2}\).

This value also satisfies the condition \(r < 1\), because if \(r^2 = \frac{\sqrt{5} - 1}{2} \approx 0.618\), then \(r = \sqrt{\frac{\sqrt{5} - 1}{2}} \approx \sqrt{0.618} \approx 0.786\), which is indeed less than 1. The result gives us the specific Value of r squared.

Comparing the Value of r² with Options

Let's check which of the provided options matches our calculated Value of r squared for the Right Angle Triangle Sides:

The calculated value matches Option 1.

Summary: Finding r² for Right Angle Triangle Sides

To determine the Value of r squared for Right Angle Triangle Sides given as \(a, ar, ar^2\) with \(r < 1\) in Geometric Progression, we used the Pythagorean Theorem. This involved setting up the equation \( (ar^2)^2 + (ar)^2 = a^2 \). Simplifying this led to a Quadratic Equation \(r^4 + r^2 - 1 = 0\) in terms of \(r^2\). Solving this equation using the quadratic formula and considering that \(r^2\) must be non-negative gives the unique valid solution \(r^2 = \frac{\sqrt{5} - 1}{2}\).