Lines Parallel to the Base

Section 1 of 2

A Line Parallel to the Base

Splits the sides in the same ratio

A line drawn parallel to the base of a triangle cuts the two sides in the same ratio — it makes a smaller triangle similar to the whole one.

Worked example — find the parallel segment

A triangle has a line parallel to its base. The top-left piece is $2$, the left side is $5$, the base is $12$. Find the parallel line $x$.

$\dfrac{x}{12} = \dfrac{2}{5}$

$x = \dfrac{2(12)}{5} = \dfrac{24}{5} = 4.8$

$x = 4.8$

Worked example — find a side piece

A triangle with a parallel line has upper sides $6$ (left) and $12$ (right along the top), and the matching lower-right piece is $4$. Find the lower-left piece $x$.

$\dfrac{x}{6} = \dfrac{4}{12}$

$x = \dfrac{4(6)}{12} = 2$

$x = 2$

Section 2 of 2

More Practice & an Exam Question

You try

A parallel line gives $\dfrac{12}{5} = \dfrac{20}{x}$. Find $x$.

Cross-multiply: $12x = 5 \times 20$.

$\dfrac{x}{20} = \dfrac{5}{12} \Rightarrow x = \dfrac{5(20)}{12} = \dfrac{100}{12} = 8.3$

$x = 8.3$

You try

A parallel line gives $\dfrac{x}{15} = \dfrac{3}{18}$. Find $x$.

$x = \dfrac{3(15)}{18}$.

$x = \dfrac{45}{18} = 2.5$

$x = 2.5$

You try

$[om]$ is parallel to $[pq]$. $|op| = 10\,\text{cm}$, $|pn| = 20\,\text{cm}$ and $|mn| = 42\,\text{cm}$. Find $|qm|$.

The small triangle (at $n$) is similar to the big one. $\dfrac{|qm|}{42} = \dfrac{10}{30}$ (since $|on| = 10 + 20 = 30$).

$\dfrac{x}{42} = \dfrac{10}{30}$

$x = \dfrac{10(42)}{30} = 14\,\text{cm}$

$|qm| = 14\,\text{cm}$

That’s Geometry — the last class!

A line parallel to the base splits the sides in the same ratio. You’ve covered angles, triangles, Pythagoras, circles, enlargements and similar triangles. Climb the rest of the pyramid to lock it in.