Page 634 1) b ≈ 2.5204, c ≈ 6.3224, C = 110° 3) b ≈ 2.1533, c ≈ 6.8184, A = 14° 5) a ≈ 17.2642, c ≈ 12.8376, A = 88° 7) b ≈ 9.6717, c ≈ 10.8509, C ≈ 41.5° 17) No Solution, b < c 19) A1 ≈ 55.1650°, C1 ≈ 104.8350°, c ≈ 14.1340 A2 ≈ 124.8350°, C2 ≈ 35.1650°, c ≈ 8.4196 21) No Solution, a < c 25) b ≈ 14.7324, c ≈ 15.2213, C = 72° 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Pre-Calculus AB PreAP/Dual, Revised ©2014 Area of a Triangle Section 10.3 Pre-Calculus AB PreAP/Dual, Revised ©2014 Viet.j.dang@lonestar.edu 2/24/2019 9:42 AM 10.3 - Area of a Triangle

The Area of a Triangle The area of any triangle is given by A = ½bh, where b is its base and h is its height. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

The Area of a Triangle The area of any triangle is given by A = ½bh, where b is its base and h is its height. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Steps in Solving Area of a Triangle Use this method in SAS Triangles Draw a picture of triangle Plug into the equation, A = ½ ac sin B The calculator is in radian mode Round to four decimal places Label appropriately 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 1 The area of triangle ABC if B = 55°, a = 34 feet, and c = 42 feet. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 2 The area of triangle ABC if a = 10 feet, b = 6 feet, and A = 42° 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 2 The area of triangle ABC if a = 10 feet, b = 6 feet, and A = 42° 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Your Turn The area of triangle ABC if B = 27°, a = 4 meters, and b = 8 meters. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Question Determine the area of this triangle in centimeters 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Question Determine the area of this triangle in centimeters 2/24/2019 9:42 AM 10.3 - Area of a Triangle

About Heron Determine the area of this triangle in centimeters Physicist, mathematician, and engineer Taught at the museum in Alexandria Interests were more practical (mechanics, engineering, measurement) than theoretical He is placed somewhere around 75 A.D. (±150) Discovered areas of quadrilaterals, polygons from 3-12 sides, and circle and surface areas of 3-D objects Heron’s Aeolipile was the first recorded steam engine. It was taken as being a toy but could have possibly caused an industrial revolution 2000 years before the original. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Proof Using Euclid’s Elements of Geometry: The opposite angles of a cyclic quadrilateral sum to two right angles all derived from the Law of Cosines 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Steps for Heron’s Formula Applied to SSS triangles Identify the semi-perimeter: 𝒔= 𝒂+𝒃+𝒄 𝟐 To find area, use the equation, 𝑨= 𝒔 𝒔−𝒂 𝒔−𝒃 𝒔−𝒄 Label correctly 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 3 Find the area of the triangle whose sides have the length of 17 ft, 25 ft, and 26 ft 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 3 Find the area of the triangle whose sides have the length of 17 ft, 25 ft, and 26 ft 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Example 4 The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of the earth.) 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Your Turn The distance from Springfield to Shelbyville is 43 miles, Shelbyville to Capital City is 53 miles, and Capital City back to Springfield is 43 miles. Eventually, if you draw a shape, it makes an oblique triangle. Determine the area. 2/24/2019 9:42 AM 10.3 - Area of a Triangle

Assignment Page 634 9-16 all 2/24/2019 9:42 AM 10.3 - Area of a Triangle