Similar Triangles

Section 1 of 2

Same shape, different size

Two triangles are similar if their corresponding angles are equal. Then the ratios of matching sides are equal: $\dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r}$.

Worked example — matching the sides

Two similar triangles. The first has sides $5$, $3$ and base $x$; the second has matching sides $8$, $y$ and base $10$. Find $x$ and $y$.

Matching the $5$ with the $8$: $\dfrac{x}{10} = \dfrac{5}{8} \Rightarrow x = \dfrac{5(10)}{8} = 6.25$

$\dfrac{y}{3} = \dfrac{8}{5} \Rightarrow y = \dfrac{8(3)}{5}\ \text{i.e.}\ \dfrac{24}{5} = 4.8$

$x = 6.25,\ y = 4.8$

Section 2 of 2

Similar Triangles in Real Life

Shadows and heights

A tall object and its shadow make a triangle similar to a short object and its shadow. Set the ratios equal.

Worked example — height from a shadow

A $1.6\,\text{m}$ stick casts a $2\,\text{m}$ shadow. At the same time a building casts a $15\,\text{m}$ shadow. Find the height $x$ of the building.

$\dfrac{x}{1.6} = \dfrac{15}{2}$

$x = \dfrac{1.6 \times 15}{2} = 12\,\text{m}$

$x = 12\,\text{m}$

Worked example — a triangle inside a triangle

A large triangle contains a smaller similar triangle. Matching sides: the small triangle has $8$ where the large has $x$, and the small base is $6$ where the large base is $20$. Find $x$.

$\dfrac{x}{8} = \dfrac{20}{6}$

$x = \dfrac{20(8)}{6} = 26.6$

$x = 26.6$

You try

Two similar triangles: the first has sides $x$, $8$, base $9$; the second has matching sides $12$, $14$, base $y$. Find $x$ and $y$.

$\dfrac{x}{14} = \dfrac{8}{12}$ and $\dfrac{y}{9} = \dfrac{12}{8}$.

$x = \dfrac{8(14)}{12} = 9.3$

$y = \dfrac{12(9)}{8} = 13.5$

$x = 9.3,\ y = 13.5$

That’s Class 10.

Similar triangles: equal angles ⇒ equal side ratios. Class 11: a line drawn parallel to the base.