The diagram shows part of the trapezium after the semi-circle was cut. AB = 26.8cm, BC = 14.4cm and diameter DC = 16cm. (i) Show that AD = 18cm. (ii) Calculate the perimeter of the shaded region.

Here's how to solve the problem:

Part (i): Show that $AD = 18$ cm.

Step 1: Draw a perpendicular line from point $D$ to $AB$. Let $E$ be the point on $AB$ such that $DE$ is perpendicular to $AB$. This forms a rectangle $EBCD$ and a right-angled triangle $ADE$.

Step 2: Determine the lengths of $DE$ and $AE$. Since $EBCD$ is a rectangle, $DE = BC = 14.4$ cm and $EB = DC = 16$ cm. Then, $AE = AB - EB = 26.8 - 16 = 10.8$ cm.

Step 3: Use the Pythagorean theorem in triangle $ADE$. In the right-angled triangle $ADE$: $AD^2 = AE^2 + DE^2$ $AD^2 = (10.8)^2 + (14.4)^2$ $AD^2 = 116.64 + 207.36$ $AD^2 = 324$ $AD = \sqrt{324}$ $AD = 18 \text{ cm}$ This shows that $AD = 18$ cm.

Part (ii): Calculate the perimeter of the shaded region.

Step 1: Identify all segments forming the perimeter. The perimeter of the shaded region consists of the lengths $AB$, $BC$, $AD$, and the arc length of the semi-circle.

Step 2: List the known lengths. $AB = 26.8$ cm $BC = 14.4$ cm $AD = 18$ cm (from part i)

Step 3: Calculate the arc length of the semi-circle. The diameter of the semi-circle is $DC = 16$ cm. The radius $r = \frac{16}{2} = 8$ cm. The arc length of a semi-circle is given by the formula $\pi r$. $\text{Arc length} = \pi \times 8 = 8\pi \text{ cm}$

Step 4: Calculate the total perimeter. Add all the lengths together: $\text{Perimeter} = AB + BC + AD + \text{Arc length}$ $\text{Perimeter} = 26.8 + 14.4 + 18 + 8\pi$ $\text{Perimeter} = 59.2 + 8\pi$ Using the value of $\pi \approx 3.14159$: $\text{Perimeter} \approx 59.2 + 8 \times 3.14159$ $\text{Perimeter} \approx 59.2 + 25.13272$ $\text{Perimeter} \approx 84.33272$ Rounding to one decimal place: $\text{Perimeter} \approx 84.3 \text{ cm}$

The perimeter of the shaded region is $\boxed{84.3 cm}$.

Send me the next one 📸