Question: "Solve for x, y, and radius of the..."

krsteva698

Answered question

2022-03-02

Solve for x, y, and radius of the arc in meters using properties of triangles, circles, and arcs.

Answer & Explanation

Athena Hussain

Beginner2022-03-03Added 5 answers

Step 1

C is the centre of the circle, part of which is shown here.
$∠ A F B = 180^{\circ} - {36.50}^{\circ} = {143.50}^{\circ}$ Linear pair property
Now, $∠ A + ∠ F + ∠ B + ∠ C = 360^{\circ}$
Sum of all angles of a quadrilateral is $360^{\circ}$
Or, $90^{\circ} + {143.50}^{\circ} + 90^{\circ} + 2 θ = 360^{\circ}$
Or, ${323.50}^{\circ} + 2 θ = 360^{\circ}$
Or, $2 θ = {36.5}^{\circ}$
Or, $θ = {18.25}^{\circ}$
Now. Let R be the radius of the circle.
In triangle FAC:
$\cos θ = \frac{R}{F C}$
Or, $F C = R \sec θ$
Now, $F E = F C - E C$

Or, $12.02 = R \sec θ - R$
Or, $R (\sec θ - 1) = 12.02$
Or, $R = \frac{12.02}{\sec θ - 1} = \frac{12.02}{\sec (18.25) - 1} = 226.94 m$
In triangle ADC:
$\sin θ = \frac{\frac{x}{2}}{R}$
Or, $x = R \sin θ = 2 (226.94) \sin ({18.25}^{\circ}) = 142.14 m$
and $\cos θ = \frac{D C}{R}$
Or, $D C = R \cos θ = 226.94 c o ({18.25}^{\circ}) = 215.53 m$
So, $y = R - D C = 226.94 - 215.53 = 11.41 m$
So, $x = 142.14 m, y = 11.41 m$ and Radius, $R = 226.94 m$

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