# Calling geniuses!!!!!!!!?

The equations y 2, y = 4x + 2, and y = -2x+ 10 form the sides of a triangle. a) Graph the triangle, and determine the coordinates of the vertices. b) Calculate the area of the triangle.

Φ² = Φ+1: The lines we have

The lines we have: y = 2, y = 4x + 2, and y = -2x + 10 Rewitten as: 0x + y = 2, 4x - y = -2, and 2x + y = 10 a) Intersect at the veritices: ((2×1-1×10)/(0×1-1×2), (0×10-2×2)/(0×1-1×2)) = (4,2), ((-2×1--1×2)/(4×1--1×0), (4×2--2×0)/(4×1--1×0)) = (0,2), and ((10×-1-1×-2)/(2×-1-1×4), (2×-2-10×4)/(2×-1-1×4)) = (4/3,22/3) b) Area is: ½| 4(2-22/3) + 0(22/3-2) + 4/3(2-2) | = ½| 4(-16/3) | = 32/3 This method does not rely on one side being parallel to an axis to simplify(?) things.

MyRank

y = 4x + 2 y = -2x + 10 y - 4x - 2 = 0 y + 2x - 10 = 0 ------------------- -6x + 8 = 0 x = +4/3 y = -2 (+4/3) + 10 y = -8/3 + 10 y = 22/3 (x, y) = (4/3, 22/3) y = 2 2 = 4x + 2 4x = 0 (x, y) = (0, 2) y = 2 2 = -2x + 10 2x = 8 x = 4 (x, y) = (4, 2) Area of triangle = ½ |x₁ x₂ x₃ x₁| |y₁ y₂ y₃ y₁| = ½ |4/3 0 4 4/3| |22/3 2 2 22/3| = ½ |(8/3 - 0) + (0 - 8) + (88/3 - 8/3)| = ½ |8/3 - 8 + 80/3| = ½ |64/3| = 32/3.

ted s

( 0 , 2 ) , ( -4 , 2 ) , & ( 4 / 3 , 22 / 3 )...base is 6 and height is 16/3

The Gnostic

Not everyone has the ability to draw graphs in our answers. I certainly do not. I can calculate the vertices this way: The first two equations coincide at 2 = 4x + 2 → 4x = 0 → x = 0. This is at point (0, 2). The first and third equations coincide at 2 = -2x + 10 → 2x = 8 → x = 4. This is at point (4, 2). The second and third equations coincide at 4x + 2 = -2x + 10 → 6x = 8 → x = 4/3. y = 4(4/3) + 2 = 16/3 + 2 = 16/3 + 6/3 = 22/3. This is at point (4/3, 22/3). The area is relatively easy: "y = 2" is a horizontal line. The point (4/3, 22/3) is 16/3 away from it since its vertical height is perpendicular to the horizontal. The area is thus ½bh = (1/2)(4)(16/3) = 32/3 square units. (The length of the base is the distance from (0, 2) to (4, 2).)