To determine the locus of a point inside a square PQRS that is equidistant from the sides PQ and QR, we need to analyze the geometric properties of the square and the concept of distance from a point to a line. ### Step-by-Step Explanation 1. **Understanding the Square**: - Let's denote the vertices of the square as follows: - \( P(0, 0) \) - \( Q(1, 0) \) - \( R(1, 1) \) - \( S(0, 1) \) - The sides of the square are: - \( PQ \) (bottom side) - \( QR \) (right side) - \( RS \) (top side) - \( SP \) (left side) 2. **Defining the Distance**: - The distance from a point \( (x, y) \) to a line can be calculated using the formula for the distance from a point to a line in the form \( Ax + By + C = 0 \). - For side \( PQ \) (the line \( y = 0 \)), the distance from a point \( (x, y) \) is simply \( |y| \). - For side \( QR \) (the line \( x = 1 \)), the distance from a point \( (x, y) \) is \( |x - 1| \). 3. **Setting Up the Equidistance Condition**: - We want to find points \( (x, y) \) such that the distance to \( PQ \) is equal to the distance to \( QR \): \[ |y| = |x - 1| \] - Since we are considering points inside the square, \( y \) will be non-negative (as it lies between 0 and 1), so we can drop the absolute value: \[ y = 1 - x \] 4. **Finding the Locus**: - The equation \( y = 1 - x \) describes a line with a negative slope that intersects the y-axis at \( (0, 1) \) and the x-axis at \( (1, 0) \). - This line is the diagonal of the square that runs from point \( Q(1, 0) \) to point \( S(0, 1) \). 5. **Identifying the Correct Option**: - The locus of points that are equidistant from sides \( PQ \) and \( QR \) is indeed the line segment \( QS \), which is the diagonal of the square. ### Evaluating the Options - **Option A: The diagonal QS** - **Correct**. This is the locus we derived. - **Option B: The perpendicular bisector of PQ** - **Incorrect**. The perpendicular bisector of \( PQ \) would be a vertical line at \( x = 0.5 \), which does not satisfy the equidistance condition from both sides. - **Option C: The diagonal PR** - **Incorrect**. The diagonal \( PR \) does not represent points equidistant from \( PQ \) and \( QR \). - **Option D: Side SR** - **Incorrect**. The side \( SR \) is not equidistant from \( PQ \) and \( QR \); it is parallel to \( PQ \) and does not satisfy the condition. ### Summary - The locus of points equidistant from sides \( PQ \) and \( QR \) is the diagonal \( QS \). - The equation derived is \( y = 1 - x \), which describes the line segment from \( Q \) to \( S \). - The other options do not satisfy the equidistance condition. ### Revision Summary - The locus of points equidistant from two lines is found by setting their distance equations equal. - For a square, analyze the distances to the relevant sides. - The correct locus in this case is the diagonal \( QS \). - Always check each option against the derived condition to confirm correctness.