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Find the area of the triang AOC in the coordinate plane

Question:

In the figure shown, the line L has the equation y = k x + b. The line L pass point C and point B. The point B has the coordinate (-4, 6) and the point C has the coordinate (0, 3). The line L intersects with the x-axis at the point A. What is the area of the triangle AOC in the coordinate plane?

How to find the area of the triangle AOC in the coordinate plane

Solution:

The area of the triangle AOC is half of the product of base and the altitude to the base. Base is segment OA. Draw a line from the vertex C and perpendicular to the x-axis, this line intersects the x-axis at point D. so the altitude to the base is the segment CD.

Solution of finding the area of the triangle AOC in the coordinate plane.

Area of triangle AOC = (1/2) b × h = (1/2) OA × h = (1/2) OA × CD

Because point C has the coordinate (-4, 6). The y-coordinate of the point C is 6. So, h = CD = 6

We need to find the length of the segment OA that is the x-coordinate of the point A.

Because point B and point C lie on the line L, so point B and point C satisfy the line equation.

Substitute the coordinate of the point C into the line equation, we get

6 = k (-4) + b ... name this as equation1

Substitute the coordinate of the point B into the line equation, we get

3 = k (0) + b ... name this as equation2

From equation2, we get b = 3. Substitute b = 3 into equation1,

6 = -4k + 3

Move the variable term -4k to the left side of the equation and move the constant term 6 to the right side of the equation.

4k = 3 - 6 = -3
k = -3/4, the slope of the line L is -3/4
The equation of the line L is y = -(3/4) x + 3

Because the line L intersects the x-axis at point A, so the point A has the coordinate (x, 0).

Substitute the coordinate of the point A into the line equation, we get

0 = -(3/4) x + 3

Move the variable term into the left side of the equation

(3/4) x = 3

Both side of the equation divide by 3

(1/4) x = 1

Both side of the equation times 4

x = 4
OA = 4
The area of the triangle AOC = (1/2) × OA × CD = (1/2) × 4 × 6 = 12

Therefore, the area of the triangle AOC is 12. Please watch the video for more details.