Let `triangleABC` be a given triangle. If `|vec(BA)-tvec(BC)|ge|vec(AC)|` for any `t in R`,then `triangleABC` is

Options

A. Equilateral
B. Right angled
C. Isosceles
D. None of these

Explanation

To solve the problem, we need to analyze the given inequality involving the vectors in triangle \( ABC \). We are given that:

\[ |\vec{BA} - t\vec{BC}| \geq |\vec{AC}| \]

for any \( t \in \mathbb{R} \). We want to determine the nature of triangle \( ABC \).

### Step 1: Rewrite the vectors
First, we can express the vectors in terms of their components. The vector \( \vec{BA} \) can be written as \( \vec{A} - \vec{B} \) and \( \vec{BC} \) as \( \vec{C} - \vec{B} \). Thus, we have:

\[ \vec{BA} - t\vec{BC} = (\vec{A} - \vec{B}) - t(\vec{C} - \vec{B}) = \vec{A} - (1 - t)\vec{B} - t\vec{C} \]

### Step 2: Analyze the magnitude
Now we need to analyze the magnitude of this vector:

\[ |\vec{BA} - t\vec{BC}| = |(\vec{A} - (1 - t)\vec{B} - t\vec{C})| \]

### Step 3: Apply the triangle inequality
Using the triangle inequality, we can state that:

\[ |\vec{BA} - t\vec{BC}| \geq |\vec{AC}| \]

This implies that the distance between points \( A \) and the line formed by points \( B \) and \( C \) is at least the length of \( AC \).

### Step 4: Consider the case when \( t = 1 \)
Let's consider the case when \( t = 1 \):

\[ |\vec{BA} - \vec{BC}| = |\vec{BA} - \vec{C} + \vec{B}| \]

This simplifies to:

\[ |\vec{BA} - \vec{BC}| = |\vec{A} - \vec{C}| \]

### Step 5: Set up the quadratic inequality
Now we can set up the quadratic inequality based on the condition given. The expression \( |\vec{BA} - t\vec{BC}| \) can be squared to yield:

\[ |\vec{BA} - t\vec{BC}|^2 \geq |\vec{AC}|^2 \]

Expanding this, we get:

\[ |\vec{BA}|^2 - 2t(\vec{BA} \cdot \vec{BC}) + t^2|\vec{BC}|^2 \geq |\vec{AC}|^2 \]

### Step 6: Formulate the discriminant
This forms a quadratic in \( t \):

\[ t^2 |\vec{BC}|^2 - 2t (\vec{BA} \cdot \vec{BC}) + (|\vec{BA}|^2 - |\vec{AC}|^2) \geq 0 \]

For this quadratic to hold for all \( t \), the discriminant must be less than or equal to zero:

\[ D = (-2(\vec{BA} \cdot \vec{BC}))^2 - 4|\vec{BC}|^2(|\vec{BA}|^2 - |\vec{AC}|^2) \leq 0 \]

### Step 7: Analyze the discriminant
This leads us to:

\[ 4(\vec{BA} \cdot \vec{BC})^2 - 4|\vec{BC}|^2(|\vec{BA}|^2 - |\vec{AC}|^2) \leq 0 \]

Dividing through by 4, we have:

\[ (\vec{BA} \cdot \vec{BC})^2 \leq |\vec{BC}|^2 (|\vec{BA}|^2 - |\vec{AC}|^2) \]

### Step 8: Conclusion
The condition implies that the angle \( C \) must be \( 90^\circ \) (since the sine of the angle must be \( 1 \)), indicating that triangle \( ABC \) is a right triangle.

### Final Answer
Thus, triangle \( ABC \) is a right-angled triangle.