Similarity and Proportion

Example 1

A triangle with sides $\displaystyle 3,4,5$ is enlarged by factor $\displaystyle 2$. Find its sides and area scale factor.

The sides are $\displaystyle 6,8,10$. The area scale factor is $\displaystyle 2^2=4$.

Example 2

A $\displaystyle 1.5$ m person casts a $\displaystyle 2$ m shadow; a tree casts an $\displaystyle 8$ m shadow at the same time. Find the tree height.

Similar right triangles give $\displaystyle h/8=1.5/2$. Thus $\displaystyle h=6$ m, assuming level ground and parallel sunlight.

Problem 3.2.1

Similar triangles have corresponding sides $\displaystyle 3$ and $\displaystyle 9$. If the smaller area is $\displaystyle 6$, find the larger area.

Show worked solution

The length factor is $\displaystyle 9/3=3$. The area factor is $\displaystyle 9$, so the larger area is $\displaystyle 54$.

Problem 3.2.2

Triangles have angles $\displaystyle 40^\circ,60^\circ,80^\circ$ and $\displaystyle 60^\circ,80^\circ,40^\circ$. Are they similar?

Problem 3.2.3

A map uses $\displaystyle 1$ cm for $\displaystyle 4$ km. What real distance corresponds to $\displaystyle 7.5$ cm?