Answer:
AB = 19.60 m and BC = 23.37 m to the nearest hundredth.
Step-by-step explanation:
The sides OA OB OC and OD are all equal and we have 2 pairs of equal isosceles triangles formed.
One with angles 80 & 2*50 and another pair with angles 100 and 2*40 degrees.
Consider the triangle with AB as the base.
Applying the Sine Rule we have:
AB / sin 80 = OA / sin 50 so AB = OA sin 80 / sin 50
Consider now the triangle BOC:
BC / sin 100 = OB / sin 50 = OA / sin 50 ( because OA = OB).
So BC = OA sin 100 / sin 40
Therefore AB /BC = (OA sin 80 / sin 50) / (OA sin 100 / sin 40)
= sin 80 * sin 40 / sin 50 sin 100
= 0.8391
and BC = AB / 0.8391.
Now we are given that the area = AB * BC = 458.
Therefore substituting for BC we have:
AB * AB / 0.8391 = 458
AB^2 = 458 * 0.8391
= 384.31
AB = 19.60
and BC = 458/19.60
= 23.37.
The length(AB) and width(BC) of the rectangle are 23.32 and 19.6
Given to us,
ABCD - rectangle
Area of ABCD = 458m2
∠AOB = 80°
We know that in a rectangle the diagonals bisect each other in two equal parts. Also, the length of the diagonals of a rectangle is equal to each other. Thus, AC = BD and AO = OC = OB =OD.
In ΔAOB,
AO = OB,
∠OAB = ∠OBA,
{Angles at the base of an isosceles triangle are equal to each other}
Also, the sum of all the angles of a triangle measures up to 180°.
∠AOB + ∠OAB + ∠OBA = 180°
80° + ∠OAB +∠OAB = 180°
2∠OAB = 180° - 80°
∠OAB = [tex]\dfrac{100}{2}[/tex]
∠OAB = 50°
In ΔABC,
We know,
[tex]\bold{Tan\theta = \dfrac{Perpendicular}{Base}}[/tex]
where,
θ = angle in degree
Perpendicular = side opposite to the angle
Base = smaller adjacent side of the angle
[tex]\bold{Tan(\angle{BCA})= \dfrac{BC}{AB}}[/tex]
[tex]\bold{Tan(50^o)= \dfrac{BC}{AB}}[/tex]
[tex]\bold{1.19= \dfrac{BC}{AB}}[/tex]
[tex]\bold{{AB}=1.19 \times {BC}}[/tex]
Area of rectangle = length x width
458 = AB x BC
458 = 1.19 x BC x BC
384.3 = BC²
BC = √384.3
BC = 19.6
Now, substituting the value of BC,
AB = 1.19 x BC
AB = 1.19 x 19.6
AB = 23.32
Hence, the length(AB) and width(BC) of the rectangle are 23.32 and 19.6 respectively.
Learn more about trigonometry:
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