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Geometry β accel. Worksheet: Area of a Triangle April 2013 I. Calculating the area of a right triangle. 1 c The area, πΎ, of a right triangle with legs π and π is given by the formula πΎ = 2 ππ. a b II. Finding the area of a triangle with SAS specified. In the triangle at right, the length of the altitude, β, can be found using the right triangle on the right: sin π₯° = β π B β = π β sin π₯° Using this relation we can derive the area, πΎ, of the triangle in terms of the two sides, π and π, and the included angle π₯°: πΎ= 1 ππ 2 h a sin π₯° xο° III. Finding the area of a triangle with SSS specified (Heronβs Formula). A C b This famous formula is credited to Heron of Alexandria and a proof can be found in his book, Metrica, written c. A.D. 60. It was probably known to the famous mathematician and philosopher Archimedes. A modern proof uses the Law of Cosines and will not be presented here, but can be found on Wikipedia. Given the three sides of a triangle, π, π, and π, the area, πΎ, of the triangle can be found using Heronβs Formula (sometimes called Heroβs Formula): πΎ = βπ (π β π)(π β π)(π β π) where π , the semiperimeter, is given by π = π+π+π 2 1. Find the area of each of the triangles below using any appropriate method. 36° 38° 14 22 12 11 15 40 32 78° 14 30 2. Find the area of the triangle with sides 5, 12, and 13 two different ways. 3. Find the area of the triangle with sides of length 7 and 10 and included angle of 50°. 4. A triangle with area 83 has two sides of lengths 24 and 17. Find the angle between these sides to the nearest degree. 5. A triangle with area 1280 has a 77° angle with one adjacent side of length 60. Find the length of the other adjacent side to the nearest tenth. Find the area of each triangle to the nearest hundredth, given the three side lengths. 6. 5, 7 πππ 9 7. 16, 34 πππ 30 8. 15, 25 πππ 30 9. 16, 18 πππ 9 9. 3.5, 2.5 πππ 4 10. 4.2, 5.6 πππ 8.2