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?
To find the ratio of the area of triangle XBY to the area of triangle ABC, we need to compare their respective areas.
Since XY is similar to AC, we can conclude that the ratio of their side lengths is the same as the ratio of their areas. Let's denote the ratio of their side lengths as k.
We are given that BY = 2 and BC = 4. Since XY is similar to AC, we can find the value of k by comparing the corresponding sides:
BY / BC = XY / AC
Substituting the given values:
2 / 4 = XY / AC
Simplifying the equation:
1 / 2 = XY / AC
Now we have the value of k = 1/2, which represents the ratio of the sides.
The area of a triangle is calculated using the formula: Area = (base * height) / 2.
Let's denote the area of triangle XBY as A(XBY) and the area of triangle ABC as A(ABC).
Since XY is similar to AC, the ratio of their areas is equal to the square of the ratio of their sides:
A(XBY) / A(ABC) = (XY / AC)^2
Substituting the value of k:
A(XBY) / A(ABC) = (1/2)^2
Simplifying the equation:
A(XBY) / A(ABC) = 1/4
Therefore, the ratio of the area of triangle XBY to the area of triangle ABC is 1/4, or in other words, the area of XBY is one-fourth the area of ABC.
To find the ratio of the area of triangle XBY to the area of triangle ABC, we need to find the lengths of sides XY and AC.
Since XY is similar to AC, we can set up a proportion between the corresponding sides:
XY / AC = BY / BC
Plugging in the values we know, we have:
XY / AC = 2 / 4
Simplifying the right side, we get:
XY / AC = 1 / 2
Now, we can find the ratio of the areas by squaring the ratio of the sides since area is proportional to the square of the side lengths:
(area of triangle XBY) / (area of triangle ABC) = (XY / AC)^2 = (1/2)^2 = 1/4
Therefore, the ratio of the area of triangle XBY to the area of triangle ABC is 1/4.