If `varphi = (1 + sqrt5)/2` and the three sides of a triangle are 1, `sqrt varphi, varphi`, which one of the following statements is true about this triangle ?

Options

A. 1. no such triangle exists
B. 2. itt's a right angle triangle
C. 3. it's an isoceles triangle
D. 4. nothing can be said

Answer

To determine the properties of the triangle with sides \(1\), \(\sqrt{\varphi}\), and \(\varphi\), where \(\varphi = \frac{1 + \sqrt{5}}{2}\), we will follow these steps:

### Step 1: Calculate \(\varphi\) and \(\sqrt{\varphi}\)

First, we need to find the numerical values of \(\varphi\) and \(\sqrt{\varphi}\).

\[ \varphi = \frac{1 + \sqrt{5}}{2} \]

Calculating \(\sqrt{5}\):

\[ \sqrt{5} \approx 2.236 \]

Now substituting this into the equation for \(\varphi\):

\[ \varphi \approx \frac{1 + 2.236}{2} = \frac{3.236}{2} \approx 1.618 \]

Next, we calculate \(\sqrt{\varphi}\):

\[ \sqrt{\varphi} \approx \sqrt{1.618} \approx 1.272 \]

### Step 2: Identify the sides of the triangle

The sides of the triangle are approximately:

- Side 1: \(1\)
- Side 2: \(\sqrt{\varphi} \approx 1.272\)
- Side 3: \(\varphi \approx 1.618\)

### Step 3: Check if the triangle inequality holds

For a set of lengths to form a triangle, the triangle inequality must hold:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Where \(a = 1\), \(b = \sqrt{\varphi} \approx 1.272\), and \(c = \varphi \approx 1.618\).

**Check the inequalities:**

1. \(1 + 1.272 > 1.618\)
- \(2.272 > 1.618\) (True)

2. \(1 + 1.618 > 1.272\)
- \(2.618 > 1.272\) (True)

3. \(1.272 + 1.618 > 1\)
- \(2.890 > 1\) (True)

Since all inequalities hold, the sides can form a triangle.

### Step 4: Check if the triangle is a right triangle

To check if the triangle is a right triangle, we can use the Pythagorean theorem. The largest side is \(\varphi\).

We need to check if:

\[ \varphi^2 = 1^2 + (\sqrt{\varphi})^2 \]

Calculating each side:

1. \(\varphi^2 \approx (1.618)^2 \approx 2.618\)
2. \(1^2 + (\sqrt{\varphi})^2 \approx 1 + (1.272)^2 \approx 1 + 1.618 \approx 2.618\)

Since both sides are equal, the triangle is a right triangle.

### Conclusion

The triangle with sides \(1\), \(\sqrt{\varphi}\), and \(\varphi\) is a right triangle.

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