Trigonometry Question
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Let ΔABC be an acute triangle. Let the intersection of the altitude from vertex B and the opposite side, ¯AC, be point D (not labeled). Let m∠A=θ. Derive the sine formula for the area of a triangle, AΔABC=12AC ABsin(θ).
| Statement | Reason |
| 1.¯BD is an alitutde | 1.Given |
| 2.AΔABC=12AC BD | 2.Standard formula for area of a triangle |
| 3.¯BD⊥¯AC | 3. |
| 4.∠ADB is a right angle | 4.Definition of perpendicular lines |
| 5.ΔADB is a right triangle | 5.Definition of a right triangle |
| 6. | 6.Sine ratio in a right triangle |
| 7.ABsin(θ)=BD | 7.Multiplication Property of Equality |
| 8.AΔABC=12AC ABsin(θ) | 8.Substitution Property of Equality |
Grade 11 Trigonometry CCSS: HSG-SRT.D.9
Does it matter that the formula specified ∠A and not one of the other angles in the triangle?
- No. Any angle in an acute triangle can be chosen, with an altitude not intersecting the vertex of that angle, and the method of derivation will be the same.
- No. Although the derivation will be different depending on the angle chosen (whether its measure is the largest, middle, or smallest of the triangle).
- Yes. This formula is only valid for the two angles with the smaller measures in an acute triangle.
- Yes. This formula cannot be derived for ∠B in any manner, since the altitude intersects vertex B.