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:
$\frac{u}{14}$u14
$=$=
$\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.
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.
Outcomes
8.10.A
Generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane
8.10.D
Model the effect on linear and area measurements of dilated two-dimensional shapes
8.3.A
Generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation
8.3.B
Compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane
8.3.C
Use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation