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?
I constructed my trap ABCD, so that
AB=5, BC=4, CD=8, and DA=6
with BC and AD both having a negative slope, (leaning to the left)
I dropped perpendiculars from B and from A to meet CD at M and N respecively.
(M is on the extension of DC)
I let MC = x, then ND = 3+x
Notice that BNMA is a rectangle.
let the height BM = AN = h
two right-angled triangles:
x^2 + h^2 = 16 --> h^2 = 16-x^2
and
(3+x)^2 + h^2 = 36 --> h^2 = 36-(3+x)^2
16-x^2 = 36-(3+x)^2
16-x^2 = 36 - 9 - 6x - x^2
6x = 11
x = 11/6
then h^2 = 16 - 121/36 = 455/36
h = √455/6
now use basic trig ratios to find any angle you want
e.g. cos(angle BCM) = x/4
= (11/6)/4 = 11/24
angle BCM = 62.72°
so angle BCD = 180-62.72 = 117.28°
and angle ABC = 62.72°
Repeat for the other side of the trapezoid
Thanks!
To find the angles of a trapezium, we'll first need to determine the lengths of the diagonals.
The diagonal can be calculated using the formula:
diagonal^2 = (difference in parallel side lengths)^2 + (sum of nonparallel side lengths)^2
Let's calculate the diagonals:
For the first diagonal:
diagonal1^2 = (8cm - 5cm)^2 + (6cm + 4cm)^2
diagonal1^2 = 3^2 + 10^2
diagonal1^2 = 9 + 100
diagonal1^2 = 109
diagonal1 = √109
For the second diagonal:
diagonal2^2 = (8cm - 5cm)^2 + (6cm - 4cm)^2
diagonal2^2 = 3^2 + 2^2
diagonal2^2 = 9 + 4
diagonal2^2 = 13
diagonal2 = √13
Now that we have the lengths of the diagonals, we can find the angles using the formula:
angle = atan(length of diagonal / difference in parallel side lengths)
Let's calculate the angles:
For the first angle:
angle1 = atan(√109 / (8cm - 5cm))
For the second angle:
angle2 = atan(√13 / (8cm - 5cm))
For the third angle:
angle3 = atan(√13 / (8cm - 5cm))
For the fourth angle:
angle4 = atan(√109 / (8cm - 5cm))
Please note that "atan" represents the inverse tangent function.
To find the angles of the trapezium, we first need to understand some properties of a trapezium. In a trapezium, the opposite angles are supplementary, meaning they add up to 180 degrees.
Let's label the sides of the trapezium for better understanding. Let side A be the longer parallel side with a length of 8 cm, side B the shorter parallel side with a length of 5 cm, side C a non-parallel side with a length of 6 cm, and side D the other non-parallel side with a length of 4 cm.
To find the angles of the trapezium, we can use the following steps:
Step 1: Determine the height of the trapezium. The height is the perpendicular distance between the parallel sides. In this case, since we don't have the height directly given, we need some additional information.
Step 2: Calculate the length of the height. To do this, we can consider the trapezium as two triangles. We can draw a line from one vertex of the trapezium, parallel to the bases, to the other base, forming a right-angled triangle. In this case, let's draw a line from the vertex between sides A and C, parallel to sides B and D, to side B.
Step 3: Calculate the length of the height using the Pythagorean theorem. We can apply the Pythagorean theorem to the right-angled triangle formed by the height, side C, and the line we drew from step 2. Substituting the values, we can find the value of the height.
Step 4: Calculate the angles of the trapezium. To do this, we can use trigonometric ratios such as sine, cosine, or tangent, depending on the angle we want to find. We can use the known sides of the trapezium and the height we calculated in step 3 to find the angles.
Please note that without knowing the height or having additional information about the trapezium, we cannot determine the exact angles of the trapezium.