8.06 Applications of similar triangles

Lesson

Practice

Lesson

The sides lengths of similar shapes are in the same ratio or proportion. So once we know that two shapes are similar, we can solve any unknown side lengths by using the ratio.

Remember!

You can write the ratio of the big triangle to the little triangle or the little triangle to the big triangle. This is helpful as it means you can always have the unknown variable as the numerator.

Worked example

Question 1

Given the two triangles below are similar, find the value of $u$u using a proportion statement.

Think: Let's equate the ratios of matching sides.

Do:

$\frac{u}{14}$`u`14	$=$=	$\frac{3}{21}$321
$\frac{u}{14}$`u`14	$=$=	$\frac{1}{7}$17	(Simplify the fraction)
$u$`u`	$=$=	$\frac{1\times14}{7}$1×147	(Multiply both sides by $14$14)
$u$`u`	$=$=	$\frac{14}{7}$147	(Now let's simplify)
$u$`u`	$=$=	$2$2	(Keep going!)

Practice questions

Question 2

Council has designed plans for a triangular courtyard in the town square.

The drawing shows the courtyard to have dimensions of $4$4 cm, $6$6 cm and $9$9 cm.

The shortest side of the actual courtyard is to be $80$80 meters long.

State the longest side length of the actual courtyard.
State the middle side length of the actual courtyard in meters.

Question 3

The two quadrilaterals in the diagram are similar.

If $a=28$a=28 m, $p=19$p=19 m and $c=17$c=17 m, solve for the exact value of $r$r.

Two similar quadrilaterals are outlined in magenta with their sides labeled. The quadrilateral on the left is larger. The top side of the larger quadrilateral is labeled $a$`a`. The left side of the larger quadrilateral is labeled $b$`b`. The bottom side of the larger quadrilateral is labeled $c$`c`. The slanted right side of the larger quadrilateral is labeled $d$`d`. The quadrilateral on the right is smaller. The top side of the smaller quadrilateral is labeled $r$`r`. The left side of the smaller quadrilateral is labeled $q$`q`. The bottom side of the smaller quadrilateral is labeled $p$`p`. The slanted right side of the smaller quadrilateral is labeled $s$`s`. The side $a$`a` of the larger quadrilateral and the side $p$`p` of the smaller quadrilateral are corresponding. The side $c$`c` of the larger quadrilateral and the side $r$`r` of the smaller quadrilateral are corresponding.

question 4

A $4.9$4.9 m high flagpole casts a shadow of $4.5$4.5 m. At the same time, the shadow of a nearby building falls at the same point (S). The shadow cast by the building measures $13.5$13.5 m. Find $h$h, the height of the building, using a proportion statement.

A building is situated on the left, casting a shadow that extends to a point marked as $S$S on the right. Adjacent to the building, a flagpole stands, casting a shadow that meets the building's shadow at point $S$S. This configuration forms two proportional triangles: one larger, delineated by the building and its shadow, and one smaller, outlined by the flagpole and its shadow. The base of the building to point $S$S measures $13.5$13.5 meters horizontally, while the flagpole, measuring $4.9$4.9 meters in height, is positioned $4.5$4.5 meters away from point $S$S along the horizontal plane. The height of the building corresponds to the height of the flagpole, and the position of the building from point $S$S corresponds to the position of the flagpole from point $S$S. The sides of the same triangle do not correspond to each other, refrain stating that $4.9$4.9 corresponds to $4.5$4.5.

Outcomes

II.G.SRT.5

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.