Applying similar triangles

Proving triangles similar, nested triangles with a parallel side, and solving shadow and height problems.

Nested triangles

If $DE \parallel BC$ in $\triangle ABC$, then $\triangle ADE \sim \triangle ABC$ (AAA: $\angle A$ is common, and corresponding angles on parallel lines are equal).

Example 1●●●

In $\triangle ABC$, $D$ is on $AB$ and $E$ on $AC$ with $DE \parallel BC$. $AD = 4$, $DB = 6$ and $DE = 5$. Find $BC$.

$AB = 4 + 6 = 10$, so $k = \dfrac{10}{4} = 2.5$.

$BC = 5 \times 2.5 = 12.5$

Shadows and heights

Example 2●●

A $1.8$ m person casts a $2.4$ m shadow. At the same time a tree casts a $16$ m shadow. How tall is the tree?

$\dfrac{h}{16} = \dfrac{1.8}{2.4}$

$h = 16 \times 0.75 = 12$ m

Flashcards

$DE parallel BC$

$\triangle ADE \sim \triangle ABC$

Nested triangles

Use the whole side of the large triangle.

Shadows

Sun's rays make equal angles, so the triangles are similar.

Practice

$DE \parallel BC$, $AD = 4$, $DB = 6$, $DE = 5$. Find $BC$.
$1.8$ m person, $2.4$ m shadow. Tree shadow $16$ m. Tree height?
$DE \parallel BC$, $AD = 3$, $AB = 9$, $BC = 18$. Find $DE$.
Why are $\triangle ADE$ and $\triangle ABC$ similar when $DE \parallel BC$?
$DE \parallel BC$, $AD = 2$, $DE = 3$, $BC = 15$. Find $AB$.
Find $DB$ in the previous question.
A $3$ m pole casts a $4$ m shadow. A building casts a $60$ m shadow. Building height?
A $2$ m stick casts a shadow of $x$ m while a $15$ m flagpole casts an $18$ m shadow. Find $x$.