Respuesta :
Answer:
smaller base is 17.21 feet
Diagonal = 18.42 feet
Explanation:
suppose, x be the shorter base.
Now, drop the heights from shorter to longer base.
thus,
Two congruent right triangles are formed
and since the trapezoid is isosceles, the heights are the same from both ends of the shorter base.
Now, it is given that the sum of base angles is 140° and the trapezoid is isosceles. thus, the base angles at both the ends will be 70°
Therefore,
we get the height as 7 × sin 70° = 6.57
now,
The base of each right triangle is (22 - x)/2 = 11 - x/2
Now, applying the concept of Pythagoras theorem, we have
(11 - x/2)²+ (7×sin70°)² = 7²
or
(11-x/2)² + 49×(sin70°)² = 49
or
(11 - x/2)² = 49 - 49×(sin70°)²
(11 - x/2)² = 5.73
or
(11 - x/2) = 2.39 and (11 - x/2) = -2.39
or
x = 17.21 and x = 26.78
since the shorter base cannot be more than 22 feet
Thus, the smaller base is 17.21 feet
also,
The base of each right triangle is 11 - x/2 = 11 - (17.21/2) = 2.39 feet
Now,the length of the diagonal can be found out by
diagonal² = (17.21)² + (7×sin70°)²
or
diagonal² =339.45
or
diagonal = 18.42 feet
(a) The length of the diagonal of the isosceles trapezoid is 20.67 ft.
(b) The length of the shorter base is 15.47 ft.
Length of the diagonal of the isosceles trapezoid
The length of the diagonal of the isosceles trapezoid is calculated as follows;
let one of the base angle = 70⁰ (since base angles is equal)
R² = a² + b² - 2bc(cosθ)
where;
- R is the length of the diagonal
- a is one of the non-parallel side
- b is base length
R² = 7² + 22² - 2(22 x 7)(cos70)
R² = 427.66
R = √427.66
R = 20.67 ft
The length of the shorter base
Let the diagonal line, bisect the one the base angle at 35 degrees.
B² = R² + a² - 2Ra(cosθ)
B² = 20.67² + 7² - (2 x 20.67 x 7 x cos35)
B² = 239.2
B = √239.2
B = 15.47 ft
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