8. Right Triangles & Trigonometry

Lesson

Recall that you can find the area of a triangle by using the formula $A=\frac{1}{2}bh$`A`=12`b``h`, where $b$`b`represents a side of the triangle, called the base, and $h$`h` represents the height of the triangle perpendicular to that side. Note that the base can be any side of the triangle as long as the height is drawn perpendicular to it.

Let's suppose we know the lengths of the sides of the triangle, but not the measurement of the height. There is another way to find the area of the triangle.

Use the applet below and the questions that follow to derive a new formula for the area of a triangle.

- Drag the vertices $A$
`A`, $B$`B`, and $C$`C`of the triangle to make the triangle larger or smaller. - Drag the "construct altitude" slider all the way to the right.
- Answer the questions below to derive the new formula for the area of a triangle.

1. Is the height of $\Delta ABC$Δ`A``B``C` the same as the height of the yellow triangle? Is it the same as the height of the pink triangle? Explain why or why not.

2. What equation relates the height of the yellow triangle to $\angle B$∠`B` and side $c$`c`? Check that you are correct by clicking the box next to "Height of Yellow Triangle".

3. Explain why the area of $\Delta ABC$Δ`A``B``C` can be represented by the equation $A=\frac{1}{2}ac\sin B$`A`=12`a``c``s``i``n``B`.

4. Find the height of the pink triangle in terms of $\angle C$∠`C` and side $b$`b`. Check that you are correct by clicking the box next to "Height of Pink Triangle".

5. Explain why the area of $\Delta ABC$Δ`A``B``C` can also be represented by the equation $A=\frac{1}{2}ab\sin C$`A`=12`a``b``s``i``n``C`.

6. It is also true that the area of $\Delta ABC$Δ`A``B``C` can be represented by the equation $A=\frac{1}{2}bc\sin A$`A`=12`b``c``s``i``n``A`. Explain why this is also true. (Hint: Would it matter if the triangle were rotated?)

(Adapted from https://www.geogebra.org/m/teZeAxWN)

Area of a triangle

$Area=\frac{1}{2}ab\sin C$`A``r``e``a`=12`a``b``s``i``n``C`

Where $a$`a` and $b$`b` are the known side lengths, and $C$`C` is the given angle between them, as per the diagram above.

Calculate the area of the following triangle.

Round your answer to two decimal places.

Calculate the area of the triangle.

Round your answer to two decimal places.

Calculate the area of the following triangle.

Round your answer to the nearest square centimeter.

Justify and apply the formula A = 1/2 ab sin (C) to find the area of any triangle by drawing an auxiliary line from vertex perpendicular to the opposite side.