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4.04 Lengths of similar figures


Problem solving using similar shapes

The sides 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 using the ratio. 


We 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 we can always have the unknown variable as the numerator.

Worked example

example 1

The following triangles are similar shapes. Find the scale factor and the value of $u$u.

Think: Compare the size of the matching sides to find the scale factor.


scale factor $=$= $\frac{3}{21}$321  
  $=$= $\frac{1}{7}$17 the triangle with $u$u is smaller, so it makes sense that the scale factor is less than $1$1.

Think: Compare the size of the matching sides to find the scale factor.


$u$u $=$= $\frac{1}{7}\times14$17×14
  $=$= $2$2


Practice questions

Question 1

In the figure below the scale factor used to enlarge the smaller quadrilateral is $4.5$4.5. Find the length of $FG$FG.

Two quadrilateral are shown. The smaller quadrilateral is named as $ABCD$ABCD while the larger quadrilateral is named as $EFGH$EFGH. In quadrilateral $ABCD$ABCD, the measures of its sides are given. Side $AD$AD measures $3$3 cm, side $CD$CD measures $4$4 cm, side $BC$BC measures $5$5 cm, and side $AB$AB is $7$7 cm.

Question 2

Consider the two similar triangles.

  1. Solve for $x$x.

  2. Solve for $c$c.

Question 3

A stick of height $1.1$1.1 m casts a shadow of length $2.2$2.2 m. At the same time, a tree casts a shadow of $6.2$6.2 m.

The tree that has $h$hm height casts a shadow of $6.2$6.2m long. At the same time, a stick with $1.1$1.1m height casts a shadow of $2.2$2.2m long. When connecting the top of the tree to the tip of its shadow, it forms a right triangle. Also, when connecting the tip of the stick to the tip of its shadow, it forms a right triangle. These two triangles formed are similar due to the side that is represented by the heights of stick and tree correspond to each other. Sides representing the lengths of their shadows are corresponding sides.


  1. If the tree has a height of $h$h metres, solve for $h$h.

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.




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