Find the area of triangle

Question

Find the area of a triangle bounded by the y axis, the line f(x)=5-1/6x and the line perpendicular to f(x) that passes through the origin

Answer 1

f(x)= 5- (1/6)x is straight line with y intercept (0,5) and slope = -1/6, a slight negative slope,

with x intercept = (30, 0)

perpendicular line has a negative inverse slope = 6, with (0,0) as the y and x intercepts if the line is through the origin

f(x) =y =6x

the two perpendicular lines intersect where 6x =5-x/6

36x =30-x

37x=30

x=30/37 and y=180/37 = 4 32/37

those two lines, and the x axis (y=0) form a right triangle with 3 vertices: (0,0), (30,0), (30/37, 180/37)

the 90 degree angle is on top at (30/37, 4 32/37)

the area is A=bh/2 where b and h are the two sloping sides on top with the hypotenuse = 30

b and h are

the distances from (30/37, 4 32/37) to the origin and from (30/37, 4 32/37) to (30,0)

calculate those two distances. multiply them, and divide by 2 to get the Area

use the distance formula a version of the Pythagorean theorem to calculate b and h

b^2 = (30/37)^2 + (180/37)^2

b=square root of sum of squares of the differences in x and y coordinates

h^2 = (30-30/37)^2 + (180/37)^2

h= square root of sum of the squares of the differences in x and y coordinates

Area =bh/2