Given angle at circumference = 30^.

Another one Alice — let's solve it. Problem 6: Step 1: Determine the angle subtended by the chord at the center. The angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the circumference. Given angle at circumference = 30^. Angle at center, = 2 × 30^ = 60^. Step 2: Use the formula for the length of a chord. The length of a chord L in a circle with radius r and subtending an angle at the center is given by the formula: L = 2r (()/(2)) Given L = 14 cm and = 60^. Step 3: Substitute the values and solve for the radius r. 14 = 2r ((60^)/(2)) 14 = 2r (30^) We know that (30^) = (1)/(2). 14 = 2r ((1)/(2)) 14 = r The radius of the circle is 14 cm. The final answer is 14 cm. Problem 7: Step 1: Identify the initial area and the transformation matrices. Initial area A_initial = 25 cm^2. First transformation matrix M_1 = 7 & 3 \\ 1 & -1 . Second transformation matrix M_2 = 2 & -3 \\ 0 & 4 . Step 2: Calculate the determinant of each transformation matrix. The area scale factor for a transformation matrix a & b \\ c & d is given by the absolute value of its determinant, |ad - bc|. For M_1: (M_1) = (7)(-1) - (3)(1) = -7 - 3 = -10 The area scale factor for M_1 is |-10| = 10. For M_2: (M_2) = (2)(4) - (-3)(0) = 8 - 0 = 8 The area scale factor for M_2 is |8| = 8. Step 3: Calculate the total area scale factor. When multiple transformations are applied sequentially, the total area scale factor is the product of the individual area scale factors. Total area scale factor = (Area scale factor for M_1) × (Area scale factor for M_2) Total area scale factor = 10 × 8 = 80. Step 4: Calculate the area of the final image. The area of the final image is the initial area multiplied by the total area scale factor. A_final = A_initial × Total area scale factor A_final = 25 cm^2 × 80 A_final = 2000 cm^2 The area of the final image is 2000 cm². The final answer is 2000 cm^2. Send me the next one 📸