`5, 5r, 5r^2 `are sides of a triangle. Which value of r cannot be possible (a) `3//2` (b) `5//4` (c) `3//4` (d) `7//4`

Options

A. `3/2`
B. `3/4`
C. `5/4`
D. `7/4`

Answer

To determine which value of \( r \) cannot be possible for the sides \( 5, 5r, 5r^2 \) of a triangle, we will apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step 1: Set up the inequalities
We have three sides:
- Side 1: \( 5 \)
- Side 2: \( 5r \)
- Side 3: \( 5r^2 \)

We will set up the following inequalities based on the triangle inequality theorem:
1. \( 5 + 5r > 5r^2 \)
2. \( 5 + 5r^2 > 5r \)
3. \( 5r + 5r^2 > 5 \)

### Step 2: Simplify the inequalities
Let's simplify each inequality one by one.

**Inequality 1:**
\[ 5 + 5r > 5r^2 \]
Dividing by 5:
\[ 1 + r > r^2 \quad \Rightarrow \quad r^2 - r - 1 < 0 \]

**Inequality 2:**
\[ 5 + 5r^2 > 5r \]
Dividing by 5:
\[ 1 + r^2 > r \quad \Rightarrow \quad r^2 - r + 1 > 0 \]
This inequality is always true for all real \( r \).

**Inequality 3:**
\[ 5r + 5r^2 > 5 \]
Dividing by 5:
\[ r + r^2 > 1 \quad \Rightarrow \quad r^2 + r - 1 > 0 \]

### Step 3: Solve the quadratic inequalities
Now we will solve the inequalities \( r^2 - r - 1 < 0 \) and \( r^2 + r - 1 > 0 \).

**For \( r^2 - r - 1 < 0 \):**
The roots of the equation \( r^2 - r - 1 = 0 \) can be found using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{5}}{2} \]
The roots are:
\[ r_1 = \frac{1 - \sqrt{5}}{2}, \quad r_2 = \frac{1 + \sqrt{5}}{2} \]
This inequality holds between the roots:
\[ \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]
Calculating the approximate values:
\[ \frac{1 - \sqrt{5}}{2} \approx -0.618, \quad \frac{1 + \sqrt{5}}{2} \approx 1.618 \]

**For \( r^2 + r - 1 > 0 \):**
The roots of the equation \( r^2 + r - 1 = 0 \) are:
\[ r = \frac{-1 \pm \sqrt{5}}{2} \]
The roots are:
\[ r_1 = \frac{-1 - \sqrt{5}}{2}, \quad r_2 = \frac{-1 + \sqrt{5}}{2} \]
This inequality holds outside the roots:
\[ r < \frac{-1 - \sqrt{5}}{2} \quad \text{or} \quad r > \frac{-1 + \sqrt{5}}{2} \]
Calculating the approximate values:
\[ \frac{-1 - \sqrt{5}}{2} \approx -1.618, \quad \frac{-1 + \sqrt{5}}{2} \approx 0.618 \]

### Step 4: Combine the results
From the first inequality, we have \( r \) must be in the range \( (-0.618, 1.618) \). From the second inequality, \( r \) must be \( r < -1.618 \) or \( r > 0.618 \).

### Step 5: Determine the possible values of \( r \)
The values of \( r \) given in the options are:
- (a) \( \frac{3}{2} = 1.5 \)
- (b) \( \frac{5}{4} = 1.25 \)
- (c) \( \frac{3}{4} = 0.75 \)
- (d) \( \frac{7}{4} = 1.75 \)

From our analysis:
- \( \frac{3}{2} \) (1.5) is valid.
- \( \frac{5}{4} \) (1.25) is valid.
- \( \frac{3}{4} \) (0.75) is valid.
- \( \frac{7}{4} \) (1.75) is **not valid** as it exceeds the upper limit of 1.618.

### Final Answer
The value of \( r \) that cannot be possible is:
**(d) \( \frac{7}{4} \)**