SOLUTION: ACDF is a Parallelogram. DEF is a sector. ABC is a right triangle. Solve for the area of the composite figure. https://ibb.co/MyW8MZ2P

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Question 1209964: ACDF is a Parallelogram. DEF is a sector. ABC is a right triangle. Solve for the area of the composite figure.
https://ibb.co/MyW8MZ2P
Found 2 solutions by CPhill, Edwin McCravy:
Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Okay, let's break down this problem step-by-step.
**1. Area of the Right Triangle ABC:**
* Since ABC is a right triangle with AB = 4 cm and BC = 4 cm, we can use the formula for the area of a triangle:
* Area = (1/2) * base * height
* Area(ABC) = (1/2) * 4 cm * 4 cm = 8 cm²
**2. Area of the Parallelogram ACDF:**
* We know that AF = 20 cm.
* Since ACDF is a parallelogram, CD = AF = 20 cm.
* To find the area of the parallelogram, we need the base and the height.
* Since the height of the parrallelogram is the same as the length of AB and BC, the height is 4cm.
* Area(ACDF) = base * height
* Area(ACDF) = 20 cm * 4 cm = 80 cm²
**3. Area of the Sector DEF:**
* We are given that angle EDF = 35 degrees.
* We need to find the radius of the sector. Since ACDF is a parallelogram, CD = AF. Also, since ABC is a right triangle with AB = BC, then the length of AC = sqrt(4^2 + 4^2) = sqrt(16+16) = sqrt(32) = 4 * sqrt(2). Because ACDF is a parallelogram, then DE = AC = 4 * sqrt(2) cm. This is the radius of the sector.
* The area of a sector is given by:
* Area = (θ/360°) * π * r²
* Area(DEF) = (35°/360°) * π * (4√2 cm)²
* Area(DEF) = (35/360) * π * 32 cm²
* Area(DEF) = (7/72) * 32π cm²
* Area(DEF) = (224/72)π cm²
* Area(DEF) = (28/9)π cm²
* Area(DEF) ≈ 9.77π cm²
* Area(DEF) ≈ 30.7 cm²
**4. Total Area of the Composite Figure:**
* Total Area = Area(ABC) + Area(ACDF) + Area(DEF)
* Total Area = 8 cm² + 80 cm² + (28/9)π cm²
* Total Area = 88 cm² + (28/9)π cm²
* Total Area ≈ 88 cm² + 30.7 cm²
* Total Area ≈ 118.7 cm²
**Therefore, the total area of the composite figure is approximately 118.7 cm².**

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor (artificial intelligence) has the wrong answer.  It probably
can't read the numbers on the drawing on the site like a human can. The drawing
on the site is off scale and on the site it looks as though if you extended AF,
it would pass through E, but this is not the case.

 

Area of parallelogram = (AB)(AF) = (4)(20) = 80 cm2

Area of right triangle ABC = expr%281%2F2%29%2AAB%2ABC=expr%281%2F2%29%2A4%2A3 = 6 cm2

To find the area of sector DEF, we must first find the radius.

The radius DF is equal to AC=sqrt%28AB%5E2%2BBC%5E2%29=sqrt%283%5E2%2B4%5E2%29=sqrt%289%2B16%29=sqrt%2825%29=5

The area of the sector is 35%2F360ths of the area of a circle with the same
radius.

So the area of sector is expr%2835%5Eo%2F360%5Eo%29%2Api%2A5%5E2=expr%28175%2F72%29picm2

So adding the parallelogram, right triangle and sector:

80+%2B+6+%2B+expr%28175%2F72%29pi+=+86%2Bexpr%28175%2F72%29pi or about 93.63581548 cm2  

Edwin