Question

on a plane are represented the points A (-4,1) B (2,1) and C (a,b) are vertices of a triangle ABC with an area equal to 24. Find the coordinates of point C

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Madelyn

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Solution:

1. Given:

- VertexA (-4,1)

- VertexB (2,1)

- VertexC (a,b)

- Area of\triangle ABC = 24

2. The formula for the area of a triangle with vertices(x_1, y_1) , (x_2, y_2) , and (x_3, y_3) :

\text{Area} = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |

3. Substitute the coordinates of pointsA and B into the formula:

\text{Area of} \ \triangle ABC = \frac{1}{2} | -4(1 - b) + 2(b - 1) + a(1 - 1) | = 24

4. Simplify the equation:

\frac{1}{2} | -4 + 4b + 2b - 2 | = 24

\frac{1}{2} | 6b - 6 | = 24

5. Solve forb :

| 6b - 6 | = 48

So,

6b - 6 = 48

6b = 54

b = 9

or

6b - 6 = -48

6b = -42

b = -7

6. The coordinates of pointC are (a, 9) or (a, -7) .

7. To finda , note that when b = 9 or b = -7 , point C lies on a line parallel to x -axis at y = 9 or y = -7 . Since point A and point B have the same y -value, any value of a would be valid for them given their symmetry on the x -axis. Therefore, point C can be (a, 9) or (a, -7) .

Thus, the coordinates of pointC are (a, 9) or (a, -7) .

The coordinates of pointC will be determined by the specific x -value of a , which can vary since the area remains consistent through the y coordinate variation given. Therefore, we depict point C in terms of a :

- Ifb = 9 , this is the valid unique coordinate for a single straight line area.

Therefore, the coordinates of pointC are (a, 9) or (a, -7)

1. Given:

- Vertex

- Vertex

- Vertex

- Area of

2. The formula for the area of a triangle with vertices

3. Substitute the coordinates of points

4. Simplify the equation:

5. Solve for

So,

or

6. The coordinates of point

7. To find

Thus, the coordinates of point

The coordinates of point

- If

Therefore, the coordinates of point

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