What is the ratio of the area of triangle XBY to the area of triangle ABC for the given measuremnts, if XYis similar to AC, and BY=2 and BC=4?
1/4
To find the ratio of the area of triangle XBY to the area of triangle ABC, we first need to find the corresponding lengths of the sides XY and AC.
Since XY is similar to AC, we know that the ratio of their corresponding side lengths is equal to the ratio of their corresponding heights (altitudes).
Let's represent the length of XY as x.
The given lengths are:
BY = 2
BC = 4
Since BY is the base of triangle XBY and BC is the base of triangle ABC, we can use their lengths to find the ratio of their altitudes.
The ratio of their altitudes is:
[h(XBY) / h(ABC)] = [BY / BC]
Given that BY = 2 and BC = 4, we have:
[h(XBY) / h(ABC)] = [2 / 4] = 1/2
So, the ratio of the altitudes of triangle XBY to triangle ABC is 1/2.
Since the ratio of the corresponding side lengths is equal to the ratio of the corresponding heights, we have:
[XY / AC] = [h(XBY) / h(ABC)]
Substituting the ratio of the altitudes we found earlier, we get:
[XY / AC] = (1/2)
Therefore, the ratio of the area of triangle XBY to the area of triangle ABC is equal to the square of the ratio of their corresponding side lengths.
[Area(XBY) / Area(ABC)] = ([XY / AC]^2) = (1/2)^2 = 1/4
So, the ratio of the area of triangle XBY to the area of triangle ABC is 1/4.