Question

In the following diagram: ABCD is equilateral in which ( AB parallel DC, DC = 3 ) and ( AD = 5 ). If angle B = x, angle D = 2x, what is the length of AB?

A. ( 2√{3} )
B. ( 3√{3} )
C. ( 4√{3} )
D. ( 5√{3} )

Answer 1

To find the length of AB in the equilateral triangle ABCD, we use the fact that the sum of angles in a triangle is 180 degrees. From there, we determine the value of angle B and use the properties of parallel lines to find the length of AB.

Step-by-step explanation:

In the given diagram, we have ABCD as an equilateral triangle with AB parallel to DC, DC = 3, and AD = 5. We are given that angle B = x and angle D = 2x. To find the length of AB, we can use the fact that the sum of angles in a triangle is 180 degrees.

Since ABCD is an equilateral triangle, all angles are equal. So, angle C = 60 degrees. Using the given information, we can set up the equation:

x + 2x + 60 = 180

Combining like terms, we get:

3x + 60 = 180

Subtracting 60 from both sides, we get:

3x = 120

Dividing both sides by 3, we get:

x = 40

So, angle B = 40 degrees. Now, we can use this information to find the length of AB. Since AB is parallel to DC, we can use the properties of parallel lines to find the length of AB. Since AB and DC are corresponding sides of the triangle, they are in proportion to their corresponding angles:

AB/DC = AB/3 = angle B/angle D = 40/80 = 1/2

Cross multiplying, we get:

AB = 3 * (1/2) = 3/2 = 1.5

So, the length of AB is 1.5.