New York Geometry - 2020 Edition

6.04 Applications of similar triangles

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.

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**:

$\frac{u}{14}$u14 |
$=$= | $\frac{3}{21}$321 | |

$\frac{u}{14}$u14 |
$=$= | $\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!) |

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.

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`.

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.

Use congruence and similarity criteria for triangles to solve problems algebraically and geometrically.

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