In triangle ABC, points X and Z are on line AB and Y is on line AC such that line XY is parallel to line BC and line ZY is parallel to line XC. If AZ = 8 and ZX = 4, then what is XB?
To find the length of XB, we need to use the concept of similar triangles.
Since line XY is parallel to line BC, we can use the property of corresponding angles. This means that angle BAC is equal to angle BXY, and angle ABC is equal to angle XYC.
From this information, we can conclude that triangle ABC is similar to triangle XYC.
Let's call XB as "a". Since triangle ABC and triangle XYC are similar, we can write the proportion of their corresponding side lengths:
AB / XY = BC / YC
Substituting the given values, we have:
AB / XY = BC / YC
AB / 4 = (AB + a) / 8
Cross-multiplying gives us:
8AB = 4(AB + a)
8AB = 4AB + 4a
Simplifying this equation, we get:
4AB = 4a
AB = a
So, we have found that AB is equal to a. Therefore, XB is equal to AB.
Given that AZ = 8 and ZX = 4, we can find AB as follows:
AB = AZ - ZB
AB = 8 - 4
AB = 4
Therefore, XB is also equal to 4.