# Identify ratio of sides in right-angled triangles

Lesson

Right-angled triangle

In the above right-angled triangle $c$c is the the hypotenuse and $\angle RPQ$RPQ is $90^\circ$90°

Now if we call $\angle PQR=\theta$PQR=θ and we can identify the opposite and adjacent sides with respect to that angle, (we did this here, if you need to refresh).  So  $b$b is the opposite side, and $a$a is the adjacent side.

A ratio is a statement of a mathematical relationship between two objects, often represented as a fraction.  Various ratios of the following can be constructed from the right-angled triangle with respect to angle $\theta$θ.

$\frac{Opposite}{Adjacent}=\frac{b}{a}$OppositeAdjacent=ba

$\frac{Adjacent}{Hypotenuse}=\frac{a}{c}$AdjacentHypotenuse=ac

$\frac{Opposite}{Hypotenuse}=\frac{b}{c}$OppositeHypotenuse=bc

#### Examples

##### Question 1

Considering the angle $\theta$θ, what is the value of the ratio $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse ?

Think: First we need to identify which sides are the adjacent and hypotenuse with respect to angle theta.  I can see that $BA$BA is the hypotenuse, $AC$AC is the opposite side and $BC$BC is the adjacent.

Do: $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{BC}{AB}=\frac{5}{13}$BCAB=513

##### Question 2

Consider the angle $\theta$θ.

What is the value of the ratio $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse?

##### Question 3

Consider the angle $\theta$θ.

What is the value of the ratio $\frac{Opposite}{Adjacent}$OppositeAdjacent?

### Outcomes

#### 10P.MT2.01

Determine, through investigation, the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios