Find the number of sides of a regular polygon if an interior angle is 3x degrees and the corresponding exterior angle is (x-20) degrees.

Step 1: Use the property that the sum of an interior angle and its corresponding exterior angle of any polygon is 180^. 3x^ + (x - 20)^ = 180^ Step 2: Solve the equation for x. 4x - 20 = 180 4x = 180 + 20 4x = 200 x = (200)/(4) x = 50 Step 3: Calculate the measure of the exterior angle. Exterior angle = (x - 20)^ = (50 - 20)^ = 30^. Step 4: Find the number of sides of the polygon. For a regular polygon, the sum of all exterior angles is 360^. If n is the number of sides, then each exterior angle is (360^)/(n). 30^ = (360^)/(n) n = (360)/(30) n = 12 The number of sides of the polygon is 12. --- Step 1: Substitute x = -2 into the first equation. x + y - z = -1 -2 + y - z = -1 y - z = -1 + 2 y - z = 1 (Equation 1) Step 2: Substitute x = -2 into the second equation. x - 2y + z = -7 -2 - 2y + z = -7 -2y + z = -7 + 2 -2y + z = -5 (Equation 2) Step 3: Solve the system of two equations (Equation 1 and Equation 2) for y and z. Add Equation 1 and Equation 2: (y - z) + (-2y + z) = 1 + (-5) y - 2y - z + z = 1 - 5 -y = -4 y = 4 Step 4: Substitute the value of y into Equation 1 to find z. y - z = 1 4 - z = 1 -z = 1 - 4 -z = -3 z = 3 The values are y = 4, z = 3. --- Step 1: Understand the enlargement transformation. For an enlargement with centre (0, 0) and scale factor k, a point (x, y) is mapped to (kx, ky). Given scale factor k = 3. Step 2: Find the image of vertex P (1, 1). P' = (3 × 1, 3 × 1) = (3, 3) Step 3: Find the image of vertex Q (2, 1). Q' = (3 × 2, 3 × 1) = (6, 3) Step 4: Find the image of vertex R (2, 2). R' = (3 × 2, 3 × 2) = (6, 6) Step 5: Find the image of vertex S (1, 2). S' = (3 × 1, 3 × 2) = (3, 6) The images of the vertices are P'(3, 3), Q'(6, 3), R'(6, 6), S'(3, 6). Last free one today — make it count tomorrow, or type /upgrade for unlimited.