ABCD is a trapezoid with DC parallel to AB, AB=4, AD=4, angle A=60, angle C=45. Find DC and BC
The two triangles above are similar. If XZ = 10 cm, YZ = 15 cm, and AC = 18 cm, what is the length of BC?
Make a sketch, marking all the given data
join BD
since AB = AD = 4, the triangle is isosceles, furthermore it is easy to see that it must be equilateral, all angles are 60° and BD = 4
By alternate angles , angle BDC = 60° leaving angle CBD = 75°
Using the sine law,
BC/sin60 = 4/sin45
BC = 4sin60/sin45 = 4.8989 or appr. 4.9
find DC in the same way
Find the radius of a right circular cyilender whose volume is 3080cu.cm and height is 20cm.
v = pi r^2 h
3080 = pi * 20 * r^2
r^2 = 3080/20pi
r = ...
To find the length of DC, we can use the properties of a trapezoid and the given information.
In a trapezoid, the sides that are parallel are called bases. Therefore, DC is one of the bases of the trapezoid.
Let's consider triangle ACD. It is an isosceles triangle since AD = CD. We also know that angle ACD is 45 degrees (given angle C = 45).
To find the length of DC, we can use the trigonometric ratio of the sine function.
sin(45 degrees) = opposite / hypotenuse
In this case, the opposite side is DC, and the hypotenuse is AD.
sin(45 degrees) = DC / AD
Rearranging the equation:
DC = AD * sin(45 degrees)
DC = 4 * sin(45 degrees)
Using the trigonometric values, we know that sin(45 degrees) = sqrt(2) / 2.
DC = 4 * (sqrt(2) / 2)
DC = 2 * sqrt(2)
Therefore, the length of DC is 2 * sqrt(2).
To find the length of BC, we can use the same approach.
Let's consider triangle ABC. It is a right triangle since angle A = 90 degrees (given angle A = 60) and angle C = 45 degrees.
Using the trigonometric ratio of the cosine function:
cos(45 degrees) = adjacent / hypotenuse
In this case, the adjacent side is BC, and the hypotenuse is AB.
cos(45 degrees) = BC / AB
Rearranging the equation:
BC = AB * cos(45 degrees)
BC = 4 * cos(45 degrees)
Again, using the trigonometric values, we know that cos(45 degrees) = sqrt(2) / 2.
BC = 4 * (sqrt(2) / 2)
BC = 2 * sqrt(2)
Therefore, the length of BC is 2 * sqrt(2).