Lesson Notes By Weeks and Term v4 - SHS 2
MEASUREMENT OF TRIANGLES
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MEASUREMENT OF TRIANGLES
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 5
Grade code: 2.2.2.LI.2
Strand code: 2
Sub-strand code: 2
Content standard code: 2.2.2.CS.1
Indicator code: 2.2.2.LI.2
Theme: GEOMETRIC REASONING AND MEASUREMENT
Subtheme: MEASUREMENT OF TRIANGLES
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Triangles are fundamental shapes in our world. We see them in the architecture of buildings like the National Theatre in Accra, in the trusses supporting the roofs of our homes, and in the patterns of Kente cloth. In Core Mathematics, we used SOH CAH TOA to solve problems involving right-angled triangles. However, most triangles in the real world are not right-angled. How does a surveyor measure a large, irregular plot of land near the Volta River? How does a pilot navigate between Kumasi and Tamale while accounting for wind?
Before we begin, let's establish a standard way to label triangles. We label the vertices (corners) with capital letters (A, B, C) and the sides opposite these vertices with corresponding lowercase letters (a, b, c). The Sine Rule
The Sine Rule establishes a relationship between the sides of a triangle and the sines of their opposite angles.
Statement: For any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant. $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$ Or, for finding angles, it's often easier to use the reciprocal form: $$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$
Derivation of the Sine Rule: Consider any triangle ABC. Let's drop a perpendicular line (height, `h`) from vertex C to the side AB. Let the point where the perpendicular meets AB be D. This creates two right-angled triangles: ΔADC and ΔBDC. In ΔADC, using SOH CAH TOA: `sin A = h / b` => `h = b sin A` (Equation i) In ΔBDC, using SOH CAH TOA: `sin B = h / a` => `h = a sin B` (Equation ii) Since both expressions are equal to `h`, we can set them equal to each other: `b sin A = a sin B` Rearranging this by dividing both sides by `sin A sin B` gives us: `b / sin B = a / sin A` If we had dropped the perpendicular from vertex A to side BC, we would similarly find that `b / sin B = c / sin C`. By combining these results, we arrive at the full Sine Rule: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$