Question

Triangle ABL is an isosceles triangle in circle A with a radius of 11, PL = 16, and ∠PAL = 93°. Find the area of the circle enclosed by line PL and arc PL. Show all work and round your answer to two decimal places.

Answer 1

The area bounded by a chord and arc it intercepts is known as a segment of a circle segment of a circle

The area of the circle enclosed by line PL and arc PL is approximately 37.62 square units

The reason the above value is correct is as follows:

The given parameters in the question are;

The radius of the circle, r = 11

The length of the chord PL = 16

The measure of angle ∠PAL = 93°

Required:

The area of part of the circle enclosed by chord PL and arc PL

Solution:

The shaded area of the given circle is the minor segment of the circle enclosed by line PL and arc PL

The area of a segment of a circle is given by the following formula;

Area of segment = Area of the sector - Area of the triangle

Area of segment = Area of minor sector APL - Area of triangle APL

Area of minor sector APL:

Area of a sector = (θ/360)×π·r²

Where;

r = The radius of the circle

θ = The angle of the sector of the circle

Plugging in the the values of r and θ, we get;

Area of the minor sector APL = (93°/360°) × π × 11² ≈ 98.2 square units

Area of Triangle APL:

Area of a triangle = (1/2) × Base length × Height

Therefore;

The area of ΔAPL = (1/2) × 16 × 11 × cos(93°/2) ≈ 60.58 square units

Required shaded area enclosed by line PL and arc PL:

Therefore, the area of shaded segment enclosed by line PL and arc PL is found as follows;

Area of the required segment PL ≈ (98.2 - 60.58) square units = 37.62 square units

The area of the circle enclosed by line PL and arc PL ≈ 37.62 square units

Learn more about the finding the area of a segment can be found here:

https://brainly.com/question/22599425

Answer 2

The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.

The calculation of the area between line segment PL and circle arc PL is described below:

1) Calculation of the area of the circle arc.

2) Calculation of the area of the triangle.

3) Subtracting the area found in 2) from the area found in 1).

Step 1:

The area of a circle arc is determined by the following formula:

$A_{ca} = \frac{\alpha\cdot \pi\cdot r^{2}}{360}$ (1)

Where:

$A_{ca}$ - Area of the circle arc.

$\alpha$ - Arc angle, in sexagesimal degrees.

$r$ - Radius.

If we know that $\alpha = 93^{\circ}$ and $r = 11$, then the area of the circle arc is:

$A_{ca} = \frac{93\cdot \pi\cdot 11^{2}}{360}$

$A_{ca} \approx 98.201$

Step 2:

The area of the triangle is determined by Heron's formula:

$A_{t} = \sqrt{s\cdot (s-l)\cdot (s-r)^{2}}$ (2)

$s = \frac{l + 2\cdot r}{2}$

Where:

$A_{t}$ - Area of the triangle.

$r$ - Radius.

$l$ - Length of the line segment PL.

If we know that $l = 16$ and $r = 11$, then the area of the triangle is:

$s = \frac{16+2\cdot (11)}{2}$

$s = 19$

$A_{t} = \sqrt{19\cdot (19-16)\cdot (19-11)^{2}}$

$A_{t} \approx 60.399$

Step 3:

And the area between the line segment PL and the circle arc PL is:

$A_{s} = A_{ca}-A_{t}$

$A_{s} = 98.201 - 60.399$

$A_{s} = 37.802$

The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.