If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7,

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Question: If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the area of the triangle?

Options:

Correct Answer: 12

Solution:

Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 1/2 | 1(5-2) + 4(2-1) + 7(1-5) | = 1/2 | 3 + 4 - 28 | = 1/2 * 21 = 10.5.

If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the area of the triangle?

If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the area of the triangle?

Step 1: Identify the coordinates of the triangle's vertices. We have A(1, 1), B(4, 5), and C(7, 2).
Step 2: Assign the coordinates to variables: x1 = 1, y1 = 1 (for A), x2 = 4, y2 = 5 (for B), x3 = 7, y3 = 2 (for C).
Step 3: Use the area formula for a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3): Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |.
Step 4: Substitute the values into the formula: Area = 1/2 | 1(5-2) + 4(2-1) + 7(1-5) |.
Step 5: Calculate each part inside the absolute value: 1(5-2) = 1*3 = 3, 4(2-1) = 4*1 = 4, 7(1-5) = 7*(-4) = -28.
Step 6: Add these results together: 3 + 4 - 28 = -21.
Step 7: Take the absolute value: |-21| = 21.
Step 8: Multiply by 1/2: Area = 1/2 * 21 = 10.5.

Area of a Triangle Using Coordinates – The formula for calculating the area of a triangle given its vertex coordinates in a Cartesian plane.
Determinants in Geometry – Understanding how to apply the determinant method to find the area of a triangle using vertex coordinates.