The parallel sides of a trapezium have lenghts 5cm and 8cm..The other two sides have the lenghts 6cm and 4cm..Find the angles of the trapezium?
To find the angles of a trapezium, we need to use the properties of a trapezium.
1. First, let's label the trapezium. Assume the shorter parallel side is represented as side AB and measures 5cm, and the longer parallel side is represented as side CD and measures 8cm. The other two sides are side BC with a length of 6cm and side DA with a length of 4cm.
A _________________ B
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D|______________ C
2. As one pair of opposite sides in a trapezium is parallel, we know that angles B and C are supplementary since they are adjacent angles formed by the intersecting sides, BC and CD.
∠B + ∠C = 180 degrees
3. We can now calculate ∠B and ∠C.
Let's assume ∠B = x degrees.
∠C = 180 - x degrees
4. The other two angles, ∠A and ∠D, can be found using the properties of a trapezium.
∠A = ∠B = x degrees
∠D = ∠C = 180 - x degrees
5. Finally, we can substitute the values we know into the equation for the sum of angles in a trapezium (∠A + ∠B + ∠C + ∠D = 360 degrees) to find the value of x.
x + x + (180 - x) + (180 - x) = 360
2x + 360 - 2x = 360
Simplifying, we get 360 = 360, which is true for all values of x.
6. Therefore, the angles of the trapezium are:
∠A = ∠B = x degrees
∠C = ∠D = 180 - x degrees
Since the equation is true for all values of x, any value can be chosen for ∠A and ∠B as long as ∠C and ∠D are their supplements.
To find the angles of a trapezium, we can use the fact that the opposite angles are supplementary.
Given that the parallel sides of the trapezium have lengths 5 cm and 8 cm, and the other two sides have lengths 6 cm and 4 cm, we can label the trapezium as follows:
8 cm
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5 cm
With this information, we can start by finding the lengths of the non-parallel sides, which are the slanting sides. Using the Pythagorean theorem, we have:
For one slanting side (let's call it side a):
a^2 = (5 - 8/2)^2 + (6^2)
a^2 = (-1)^2 + 36
a^2 = 1 + 36
a^2 = 37
a = √37 cm
For the other slanting side (let's call it side b):
b^2 = (8 - 5/2)^2 + (4^2)
b^2 = (7/2)^2 + 16
b^2 = 49/4 + 64/4
b^2 = 113/4
b = √(113/4) cm
Next, we can identify the angles of the trapezium:
Angle A: This is the angle between the longer parallel side (8 cm) and the slanting side (a).
tan(A) = (6 cm) / (√37 cm)
A = arctan(6 / √37)
Angle B: This is the angle between the shorter parallel side (5 cm) and the slanting side (b).
tan(B) = (4 cm) / (√113/4 cm)
B = arctan(4 / √(113/4))
Angle C: This is the angle between the longer parallel side (8 cm) and the shorter parallel side (5 cm).
C = 180° - (A + B)
Finally, you can use a calculator or appropriate software to find the values of angles A, B, and C.
If AB=8 and DC=5, then if the altitude is h,
h = 4sinA = 6sinB
h cotA + 5 + h cotB = 8
4 sinA cotA + 5 + 6 sinB cotB = 8
4cosA + 5 + 6cosB = 8
4cosA + 6cosB = 3
solve for A and B.
Note that D = (90-A)+90 = 180-A
and similarly for C.