Trigonometry Question
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Let ΔABC be an obtuse triangle, where m∠BAC>90°. Let the intersection of the altitude from vertex B and the extension of side ¯AC be point D (not labeled), such that AD+AC=CD. Let m∠BAC=θ and m∠BAD=α. 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.α+θ=180° | 3. |
| 4.α=180°-θ | 4.Subtraction Property of Equality |
| 5.¯BD is an alitutde | 5.Previous result |
| 6.¯BD⊥¯AD | 6. |
| 7.∠ADB is a right angle | 7.Definition of perpendicular lines |
| 8.ΔADB is a right triangle | 8.Definition of a right triangle |
| 9. | 9.Sine ratio in a right triangle |
| 10.sin(180°-θ)=BDAB | 10.Substitution Property of Equality |
| 11.sin(θ)=BDAB | 11.Trig Identity |
| 12.ABsin(θ)=BD | 12.Multiplication Property of Equality |
| 13.AΔABC=12AC ABsin(θ) | 13. |
Grade 11 Trigonometry CCSS: HSG-SRT.D.9
What is the missing reason in step 3?
- Sum of an exterior angle of a triangle and any interior angle is 180°
- Sum of the angles on a straight line is 180°
- Sum of the angles of the triangle is 180°
- Exterior Angle Theorem