In Diagram, PQR is an arc of a circle with centre T. PRT is an equilateral triangle and RST is a semicircle with centre 0 . Given \( R T=5.6 \mathrm{~cm} \), find the nerimeter. in cm. of the whole diagram.

To find the perimeter of the whole diagram consisting of the arc PQR, the equilateral triangle PRT, and the semicircle RST, we need to calculate the lengths of each component. 1. **Length of the arc PQR**: Since PRT is an equilateral triangle, all sides are equal. The length of each side of triangle PRT is equal to \( RT \), which is given as \( 5.6 \, \text{cm} \). Therefore, \( PR = 5.6 \, \text{cm} \) and \( PT = 5.6 \, \text{cm} \). The angle at the center T for an equilateral triangle is \( 60^\circ \). The length of the arc can be calculated using the formula for the length of an arc: \[ \text{Arc length} = \frac{\theta}{360^\circ} \times 2\pi r \] where \( \theta \) is the angle in degrees and \( r \) is the radius (which is \( RT = 5.6 \, \text{cm} \)). \[ \text{Arc length} = \frac{60}{360} \times 2\pi \times 5.6 = \frac{1}{6} \times 2\pi \times 5.6 = \frac{5.6\pi}{3} \, \text{cm} \] 2. **Length of the sides of triangle PRT**: The perimeter of triangle PRT is: \[ \text{Perimeter of } PRT = PR + RT + PT = 5.6 + 5.6 + 5.6 = 16.8 \, \text{cm} \] 3. **Length of the semicircle RST**: The radius of the semicircle RST is \( RS = RT = 5.6 \, \text{cm} \). The length of the semicircle is given by: \[ \text{Length of semicircle} = \frac{1}{2} \times 2\pi r = \pi r = \pi \times 5.6 \, \text{cm} \] 4. **Total perimeter**: The total perimeter of the whole diagram is the sum of the lengths of the arc PQR, the sides of triangle PRT, and the semicircle RST. However, we need to be careful not to double-count the segments that are shared. The perimeter can be calculated as: \[ \text{Total perimeter} = \text{Length of arc PQR} + \text{Length of semicircle RST} + \text{one side of triangle PRT} \] Since the arc PQR and the semicircle RST share the segment RT, we only count one side of the triangle. Therefore, the total perimeter is: \[ \text{Total perimeter} = \frac{5.6\pi}{3} + \pi \times 5.6 + 5.6 \] \[ = \frac{5.6\pi}{3} + \frac{16.8\pi}{3} + 5.6 \] \[ = \frac{22.4\pi}{3} + 5.6 \] Thus, the total perimeter of the whole diagram is: \[ \text{Total perimeter} = \frac{22.4\pi}{3} + 5.6 \, \text{cm} \] This is the final expression for the perimeter in cm. If you need a numerical approximation, you can substitute \( \pi \approx 3.14 \) to get a numerical value.