Vectors: Finding Ratio Of Area
In this topic, you will learn how vectors can be used to find the ratio of areas in geometric figures.
We will cover two methods:
- Similar Triangle
- Common Height
Related Lessons:
Find Ratio Of Area Using Similar Triangles
Recall, when 2 triangles are similar, the ratio of their area is the squared of their length:
- We can use the formula \(\frac{A_1}{A_2}=\left(\frac{l_1}{l_2}\right)^2\)
- Generally, to identify two similar triangles, we look for pairs of parallel lines that create equal corresponding angles. The triangles either appear as a smaller triangle inside a larger triangle, or they are arranged in a “figure of 8” shape where the lines intersect and form two distinct triangles.
Finding Ratio Of Area Using Common Base
Another way to find the ratio of area is to equate the area of two triangles with equal height. If two triangles share the same height H, and their base length are \(B_1\) and \(B_2\) respectively, the ratio of their area is:
\[
\begin{align*}
\frac{A_1}{A_2}
&= \frac{\frac{1}{2} B_1 H}{\frac{1}{2} B_2 H} \\
&= \frac{B_1}{B_2}
\end{align*}
\]