4.03 Proportions and similar polygons

Lesson

Practice

Lesson

Recall the side lengths of similar shapes are in the same ratio or proportion. So once we know that two shapes are similar, we can find any unknown side lengths by setting up a proportion.

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 because it means you can always have the unknown variable as the numerator.

Worked examples

Question 1

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

Think: Let's set up a proportion for matching sides. To make this easier, think of it in words first: "$u$u is to $14$14 as $3$3 is to $21$21".

Do:

$\frac{u}{14}$`u`14	$=$=	$\frac{3}{21}$321	(Simplify the fraction)
	$=$=	$\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

Reflect: We now know that the length of side $u$u is $2$2 units. Let's check to make sure that is reasonable by using the scale factor. The scale factor is $7$7 because if you multiply the side of length $3$3 by a factor of $7$7 you get the corresponding side length of $21$21. So let's use that scale factor of $7$7 with side $u$u. If we multiply $2$2 by $7$7 we get the corresponding side length of $14$14 so we can be confident that our solution is correct.

Practice questions

Question 2

Find the value of $u$u using a proportion statement.

Two similar triangles. The smaller triangle, positioned on the left, has its vertices labeled A, B, and C. The larger triangle, situated on the right, has its vertices labeled D, E, and F. Angle A of triangle(ABC) is congruent to angle D of triangle(DEF), as indicated by a single arc on each of these angles. Additionally, angle C of triangle(ABC) is congruent to angle F of triangle(DEF), with this congruence denoted by two arcs on these angles. In $\triangle ABC$△`ABC`, sides AB which is opposite to angle C measures $16$16 units. Side AC which is opposite to angle B measures $8$8 units. In $\triangle DEF$△`DEF`, side DE which is opposite angle F measures $u$`u` units, and side DF which is opposite angle E measures $18$18 units

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

8.G.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.