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. A. Find the equation of the line passing through B and perpendicular to AC? B. Let the point of intersection of AC with the line you found in part A be point D. Find the coordinates of point D? C. Use the distance formula to find the length of the base and the height of ΔABC? D. Find the area of ΔABC?

The area of ΔABC is 3.5 unit²

Step-by-step explanation:

Given that the coordinates of the triangle are;

A = (2, 1), B = (3, 3), C = (1, 6)

The slope, m₁ of the line joining A to C is found as follows;

m₁ = (6 - 1)/(1 - 2)  = -5

From the general equation of a straight line, y =m·x  + c, we have;

6 = -5×1 + c

∴ c = 6 + 5 = 11

The equation of the line AC is therefore;

y = -5·x + 11

The equation of a perpendicular line to another line is the negative reciprocal of the slope of the line

Therefore, we have;

y - 3 = 1/5(x - 3)

y = 1/5x - 3/5 + 3

y = 1/5x + 12/5

5·y = x + 12

At the point of intersection, D, of the line from vertex B and the line AC, we have;

Equating both lines to each other gives;

y = -5·x + 11 =  1/5x + 12/5

1/5x + 5·x = 11 - 12/5 = 42/5

26/5·x = 42/5

x = 43/26

y = 71/26

D = (43/26, 71/26)

The length of the base AC = √((2 - 1)² + (1 - 6)²) = √26

The height of ΔABC = length of segment BD = √((43/26 - 3)²+ (71/26 -  3)²) = 7/√26

The area of triangle ABC = 1/2×AC×BD = 1/2×√26×7/√26 = 7/2 = 3.5 unit²

The area of ΔABC = 3.5 unit².

arrange the equations with like terms in columns.

analyze the coefficients of x or y.

add the equations and solve for the remaining variable.

substitute the value into either equation and solve.

check the solution.

step-by-step explanation:

step-by-step explanation: