a. If \( \mathrm{A}(-3,2), \mathrm{B}(-5,4) \) and \( \mathrm{C}(-2,6) \) are the vertices of \( \triangle \mathrm{ABC} \) in a graph and rotate it through \( 90^{\circ} \) in clockwise direction about origin. Write down the coordinates of the vertices of the image. (3) b. If the center of a circle \( \mathrm{O}(5,4) \) and \( \mathrm{A}(9,4) \) be any point its circumference, find the radius of the circle. (1)

To find the new coordinates of the vertices after a \(90^\circ\) clockwise rotation about the origin, you can use the transformation formula: If a point \( (x, y) \) is rotated \(90^\circ\) clockwise, the new coordinates will be \( (y, -x) \). Applying this to the vertices: - For \(A(-3, 2)\), the new coordinates are \( (2, 3) \). - For \(B(-5, 4)\), the new coordinates are \( (4, 5) \). - For \(C(-2, 6)\), the new coordinates are \( (6, 2) \). So, after rotation, the coordinates of the vertices are: - \( A'(2, -3) \) - \( B'(4, -5) \) - \( C'(6, -2) \) For the second part, the radius of the circle can be found as the distance from the center \(O(5, 4)\) to any point on the circumference \(A(9, 4)\). Using the distance formula: \[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points \(O\) and \(A\): \[ \text{Radius} = \sqrt{(9 - 5)^2 + (4 - 4)^2} = \sqrt{(4)^2} = 4 \] Thus, the radius of the circle is \(4\).