Pythagoras` Theorem and the Trigonometric ratios (sin, cos, tan) require right-angled triangles. Both the theorem and the trigonometric ratios can be used on more general triangles that do not contain a right angle.
Triangles can be divided into two smaller triangles:
1. A triangle ABD is shown below. A vertical line dropped from A intersects the line BD at C. The ratio of the lengths BC:CD is 2:7. What is the distance BD?
The distance AC is the same for both triangles ABC and ACD
Let the distance BC be 2y and CD be 7y
`AC = sqrt(8^2 - (2y)^2) = sqrt(12^2 - (7y)^2)`
`8^2 - 4y^2 = 12^2 - 49y^2`
`45y^2 = 80`
`y^2 = 1.7778`
`y = 1.3333`
`BD = 2y + 7y = 9y = 12`
2. What is the area of the triangle ABC, shown below?
Answer: 20.8 cm2
Draw a vertical from A. Let this be length x
sin = `frac(text(opposite))(text(hypotenuse))`
`sin 60 = frac(x)(8)`
`x = 6.928`
Area of a triangle = `frac(1)(2) xx bh`
Area = `frac(1)(2) xx 6 xx 6.928`
A= 20.785 = 20.8 to 1dp
(Or use the formula for the area of any triangle. This question derives the formula.)