Animated Solution for Mathematics - Trigonometry: If 5, 5r, $5r^2$ are the lengths of the sides of a triangle, then r cannot be equal to :

Imagine you are standing in a workshop, tasked with building a triangle using three wooden rods. You are given lengths of $5$, $5r$, and $5r^2$.

Geometry is a strict master. If you pick an $r$ that is too large or too small, those rods will never meet at the corners. They will either be too short to bridge the gap or one will be so long that the other two cannot reach its ends.

We must master the art of the Triangle Inequality Theorem to find exactly which values of $r$ are allowed.

For any three segments to form a triangle, they must satisfy the Triangle Inequality Theorem. This is the fundamental requirement for closure in Euclidean space.

We have three conditions to satisfy:

1. $5+5r>5r^2$ 2. $5r+5r^2>5$ 3. $5r^2+5>5r$

Let's simplify these. By dividing each by $5$, we obtain:

These are the gates we must pass through.

Let's look at the first condition: $r^2-r-1<0$. To solve this, we find the roots of the quadratic $r^2-r-1=0$ using the quadratic formula:

Since we know $r$ must be positive, we focus on the interval $0<r<\frac{1+\sqrt{5}}{2}$. This gives us an upper bound of approximately $1.618$.

Next, consider the second condition: $r^2+r-1>0$. We find the roots of $r^2+r-1=0$, which are $r=\frac{-1\pm \sqrt{5}}{2}$.

Since we need the expression to be greater than zero, we take the positive root: $r>\frac{\sqrt{5}-1}{2}$, which is approximately $0.618$.

Finally, consider the third condition: $r^2-r+1>0$. If you calculate the discriminant $D=(-1{)}^2-4(1)(1)=-3$, you will see it is negative.

This means the parabola $r^2-r+1$ never touches the $x$-axis. Since it opens upwards, it is always positive for any real $r$. This condition is satisfied for all values.

Now, we combine our findings. We need $r$ to be greater than $0.618$ AND less than $1.618$.

Our valid range for $r$ is:

Now, look at your options. We have $\frac{3}{2}=1.5$, $\frac{3}{4}=0.75$, and $\frac{5}{4}=1.25$. All of these fall comfortably within our 'Golden' range.

However, consider $\frac{7}{4}=1.75$. This value is clearly greater than $1.618$.

If you tried to build a triangle with $r=1.75$, the side $5r^2$ would be so long that the other two sides, $5$ and $5r$, would simply fail to connect.

Mathematics is often about finding the boundaries of the possible. By applying the Triangle Inequality, we defined the physical limits of a geometric shape.

The value $r=\frac{7}{4}$ lies outside these limits, making it the only impossible choice. Keep this logic in your toolkit—whenever you see constraints, look for the inequalities that define them.