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Name: _______________________________ Common Core Geometry – Honors Date: ________________ Area of a Triangle Using Trig AIM: How do we use trigonometry to determine the area of a triangle? Do Now: A man is walking his dog on level ground in a straight line with the dog's favorite tree. The angle of elevation from the man's present position to the top of a nearby telephone pole is 30º. The angle of elevation from the tree to the top of the telephone pole is 45º. If the telephone pole is 40 feet tall, how far is the man with the dog from the tree? Express answer to the nearest tenth of a foot. Homework: Worksheet We are all familiar with the formula for the area of a triangle, , where b stands for the base and h stands for the height drawn to that base. (the lettering used is of no importance) In the triangle at the right, the area could be expressed as: Now, let's be a bit more creative and look at the diagram again. By using the right triangle on the left side of the diagram, and our knowledge of trigonometry, we can state that: This tells us that the height, h, can be expressed as bsinC. If we substitute this new expression for the height, we can write the triangle area formula as: (where a and b are adjacent sides and C is the included angle) Example: Given the triangle below, find its area. Express the area rounded to three decimal places. Given the parallelogram shown at the right, find its EXACT area. If we are looking for an EXACT answer, we do NOT want to round our value for sin 60º. We need to remember that the sin 60º (from our 30º- 60º- 90º reference triangle) is . Now, the diagonal of a parallelogram divides the parallelogram into two congruent triangles. So the total area of the parallelogram will be double the area of one of the triangles formed by a diagonal. square units. We discovered, due to the doubling, that the area of a parallelogram is really just Parallelogram (where a and b are adjacent sides and C is the included angle) Class Work: 1. In ABC, AB = 10, AC = 8, and m<A = 45º. Find the area of ABC, to the nearest tenth of a square unit. 2. In an isosceles , the two equal sides each measure 24 meters, and they include an angle of 30º. Find the area of the isosceles triangle, to the nearest sq. meter. 3. In ABC, AB = 12 meters and AC = 20 meters. If the area of the triangle is 77 sq. meters, find the measure of <A, to the nearest degree. 4. In a rhombus, each side is 15, and one angle is 130º. Find the area of the rhombus, to the nearest square unit. 5. A farmer has a triangular field where two sides measure 450 yards and 320 yards. The angle between these two sides measures 80º. The farmer wishes to use an insecticide that costs $4.50 per 100 sq. yards or any part of 100 yds. What will it cost to use this insecticide on this field? 6. A triangle has two sides of 30 meters and 26 meters, and the angle between them is an obtuse angle. If the area of the triangle is 300 sq. meters, find the measure of the obtuse angle (to the nearest degree.)