We are given following diagram and want to find the length of the log. We will identify the vertices of the triangles with letters to make our calculations easier. Let's first show that triangles XTZ and YWZ are similar. Once we know that the triangles are similar, we can use their proportional relationship to find the missing distance.
For △ XTZ and △ YWZ to be similar, at least two of their angles need to be congruent. We can see from the angle markers shown in the diagram that ∠ X is congruent to ∠ Y and that ∠ T is congruent to ∠ W.
∠ X ≅ ∠ Y
∠ T ≅ ∠ W
Since two angles of △ XTZ are congruent to two angles of △ YWZ, they are similar by the Angle-Angle (AA) Similarity rule.
△ XTZ ~ △ YWZ
Now, notice that XT corresponds to YW and that XZ corresponds to YZ.
Corresponding sides of similar figures will have proportional lengths. We know that XT= 8 meters, WZ= 4 meters, and TZ= 12 meters, so we will use these lengths to write a proportion.
XT/YW=TZ/WZ ⇔ 8/x=12/9
Let's solve this proportion using cross products.
