Question

Finding the area of a triangle is straightforward if you know the length of the base and the height of the triangle. But is it possible to find the area of a triangle if you know only the coordinates of its vertices? In this task, you’ll find out. Consider ΔABC, whose vertices are A(2, 1), B(3, 3), and C(1, 6); let line segment AC represent the base of the triangle.Part AFind the equation of the line passing through B and perpendicular to .Part BLet the point of intersection of AC with the line you found in part A be point D. Find the coordinates of point D.

Answer 1

PART A)

We want to find the equation of the line passing through point B and perpendicular to line AC. The coordinates of line AC are A(2, 1) and C(1, 6)

The equation of a line in th slope intercept form is expressed as

y = mx + c

Where

m represents slope

c represents y intercept

We would find the slope, m by applying the formula,

m = (y2 - y1)/(x2 - x1)

Considering line AC,

x1 = 2, y1 = 1

x2 = 1, y2 = 6

m = (6 - 1)/(1 - 2) = 5/- 1

m = - 5

If two lines are perpendicular, it means that the slope of one line is the negative reciprocal of the slope of the other line. This means that the slope of the passing through B and perpendicular to AC is 1/5

Since the line passes through B whose coordinate is (3, 3), we would find the equation of the perpendicular line by substituting x = 3, y = 3 and m = 1/5 into the slope intercept equation to get the y intercept, c. Thus, we have

3 = 1/5 * 3 + c

3 = 3/5 + c

c = 3 - 3/5

c = 12/5

By substituting m = 1/5 and c = 12/5 into the slope intercept equation, the equation of the line is

y = x/5 + 12/5

PART B)