General Triangles

# General Triangles

GCSE(H),

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: If angles are given, the sine rule or the cosine rule are more likely to be used to solve problems involving triangles.

## Examples

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? 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.)