The diagram below not drawn to scale represents a plot of land in the shape of an isosceles trapezoid and a semicircle. ie: AB =CD and AE = DF. The parallel sides BC and AD are of length 11 m and 21 m respectively. The sides BE=CF=12m. a) Calculate the length of the side of the land AB. b) Calculate the distance around the entire plot of land to 3 significant figures.
The Correct Answer and Explanation is:
We can find the length of AE by dividing the combined length by two.
AE = 10 m / 2 = 5 m.
Now we have a right-angled triangle ABE with a base AE = 5 m and a height BE = 12 m. We can use the Pythagorean theorem (a² + b² = c²) to find the length of the hypotenuse, AB.
AB² = AE² + BE²
AB² = 5² + 12²
AB² = 25 + 144
AB² = 169
AB = √169
AB = 13 m
**b) Calculate the distance around theHere are the solutions to the problem.
a) Calculation for the length of side AB
To find the length of the side AB, we can focus on the right-angled triangle ABE. The length AB is the hypotenuse of this triangle. We are given the height BE = 12 m. We need to determine the length of the base AE.
The total length of the bottom side of the trapezoid is AD = 21 m. The length of the top side is BC = 11 m. Since BE and CF are perpendicular to AD, the shape BCFE is a rectangle. Therefore, the length of EF is equal to the length of BC, which is 11 m.
The length of AD is the sum of the segments AE, EF, and DF. So, AD = AE + EF + DF.
Substituting the known values:
21 m = AE + 11 m + DF
This means the combined length of AE and DF is:
AE + DF = 21 m – 11 m = 10 m
The problem states that the trapezoid is isosceles and AE = DF. Therefore, we can find the length of AE by dividing the combined length by 2:
AE = 10 m / 2 = 5 m
Now we have the lengths of the two shorter sides of the right-angled triangle ABE: AE = 5 m and BE = 12 m. We can use the Pythagorean theorem (a² entire plot of land to 3 significant figures.**
The distance around the entire plot is its perimeter. This includes the curved part of the semicircle, the side CD, the base AD, and the side AB. The side BC is an internal boundary and is not part of the total perimeter.
- Length of the semicircular arc: The formula for the circumference of a semicircle is (πd)/2, where d is the diameter. The diameter of this semicircle is the length of BC, which is 11 m.
Arc Length = (π × 11) / 2 ≈ 17.2788 m. - Length of side CD: + b² = c²) to find the hypotenuse AB.
AB² = AE² + BE²
AB² = 5² + 12²
AB² = 25 + 144
AB² = 169
AB = √169
AB = 13 m
The length of the side of the land AB is 13 m.
b) Calculation for the distance around the entire plot
The distance around the entire plot of land is its perimeter. This consists of the length of the side AB, the length of the base AD, the length of the side CD, and the length of the curved arc of the semicircle. The side BC is an Since the trapezoid is isosceles, CD = AB. From part (a), we know AB = 13 m, so CD = 13 m.
- Length of base AD: This is given as 21 m.
- Length of side AB: This was calculated in part (a) as 13 m.
Now, we add these lengths together to find the total perimeter.
Perimeter = Arc Length + CD + AD + AB
Perimeter ≈ 17.2788 m + 13 m + 21 m + 13 m
Perimeter ≈ 64.2788 m.
The question asks for the answer to 3 significant figures. The first three significant figures are 6, 4, and 2. The fourth digit is 7, which is 5 or greater, so we round up the third digit. internal line and not part of the total perimeter.
First, we calculate the length of the semicircle’s arc. The
The distance around the plot is 64.3 m. formula for the circumference of a full circle is C = πd. For a semicircle, the arc length is half of that.
Arc Length = (1/2) * π * d
The diameter (d) of the semicircle is the length of BC, which is 11 m.
Arc Length = (1/2) * π * 11 ≈ 17.279 m
Next, we sum the lengths of all the outer sides:
Perimeter = Arc Length + CD + AD + AB
Since the trapezoid is isosceles, CD = AB = 13 m. The length of AD is given as 21 m.
Perimeter ≈ 17.279 m + 13 m + 21 m + 13 m
Perimeter ≈ 64.27thumb_upthumb_down
