Question

Pleasr help me out with this ​

Answer 1

`\textsf{Perimeter}=3\pi-(\pi)/(√(2))=7.20333649...\; \sf cm`

Explanation:

`\boxed{\begin{minipage}{6.4 cm}\underline{Arc length}\\\\Arc length $=r \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}`

To convert degrees to radians, multiply the angle in degrees by π/180°.

Arc AB

As arc AB has center O, the radius of arc AB is OA = 1 cm.

As AB is a straight line, ∠AOB is 180° = π.

Therefore, the length of arc AB is:

`\implies AB=r \:\theta=1 \cdot \pi = \pi`

Arc AC

Triangle BOE is a right triangle with base of 1 cm and height of 1 cm.

Therefore, ∠OBE is 45° = π/4

If arc AC has center B, then the radius is AB = 2 cm.

Therefore, the length of arc AC is:

`\implies AC=r \:\theta=2 \cdot (\pi)/(4) = (\pi)/(2)`

Arc BD

Arc BD is the same as arc AC.

`\implies BC= (\pi)/(2)`

Arc CD

As triangle BOE is a right triangle with base of 1 cm and height of 1 cm, the length of its hypotenuse BE is:

`\implies BE=√(1^2+1^2)=√(2)`

As arc AC has center B and radius of AB = 2 cm, then BC is also its radius and therefore BC = 2 cm

Therefore:

`\implies CE=BC-BE`

`\implies CE=2-√(2)`

The arc CD has center E so its radius is CE = 2-√2.

As ∠BEO and ∠AEO are both 45° then ∠AEB is 90°.

According to the vertical angle theorem, ∠CED is also 90° = π/2.

Therefore, the length of arc CD is:

`\implies CD=r \:\theta=(2-√(2)) \cdot (\pi)/(2) = ((2-√(2))\pi)/(2)`

Perimeter of the egg

The perimeter of the egg is the sum of the found arcs:

`\implies \textsf{Perimeter}=AB+AC+BD+CD`

`\implies \textsf{Perimeter}=\pi+(\pi)/(2)+(\pi)/(2)+((2-√(2))\pi)/(2)`

`\implies \textsf{Perimeter}=2\pi+((2-√(2))\pi)/(2)`

`\implies \textsf{Perimeter}=2\pi+(2 \pi)/(2)-(√(2)\:\pi)/(2)`

`\implies \textsf{Perimeter}=3\pi-(√(2)\:\pi)/(2)`

`\implies \textsf{Perimeter}=3\pi-(\pi)/(√(2))`

`\implies \textsf{Perimeter}=7.20333649...\; \sf cm`

Answer 2

• 3π - π√2/2 or 7.2 cm

-------------------------

According to the given we can state:

• AD = BC = AB = 2AO = 2,
• EO is the perpendicular bisector of AB,
• Arc AC and arc BD are of equal length.

Find arc AB, it is a semicircle of 1 cm radius:

• arc(AB) = 1/2 × 2πr = πr = π cm

ΔAEO and ΔBEO are both isosceles, hence ∠A and ∠B are both 45°.

Find arcs ABD and BAC:

• arc(ABD) + arc(BAC) = 2 × 45/360 × 2πr = 1/2 × πr = π cm

∠AEB is a right angle since ∠AEO and ∠BEO are both 45°.

Hence ∠CED is also right angle as vertical angle with ∠AEB.

Find the length of EC and ED.

We know AD = BC = 2 cm and ΔAEO is 45° right triangle.

It gives us:

• AE = √2

Then:

• ED = AD - AE = 2 - √2

Find arc ECD:

• arc(ECD) = 1/4 × 2πr = 1/2 × π(2 - √2) = π - π√2/2 cm

The perimeter is the sum of all the arc measures:

• P =
• π + π + π - π√2/2 =
• 3π - π√2/2 or 7.2 cm (rounded)