Using similarity proportion to solve problems

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.

Examples

Question 1

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

Question 2

Consider the two similar triangles.

Solve for $x$x.
Solve for $c$c.

Question 3

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

10.G.T.1

Definitions, examples, counterexamples of similar triangles, covering (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio, (b) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side, (c) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar, (d) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar (e) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.