Praxis Core Academic Skills for Educators: Mathematics (5733)GeometryHard
A machinist is programming a robotic arm to cut a triangular piece from a sheet of metal. The vertices of the triangle are at A(1, 2), B(7, 2), and C(4, 8). What is the area of this triangle?
- A24 square units
- B12 square units
- C18 square units
- D36 square units
Show answer & explanationAnswer & explanation
Correct answer: C. 18 square units
To find the area of a triangle given its vertices, we can use the base and height method if two vertices share a coordinate. Here, A(1, 2) and B(7, 2) have the same y-coordinate, so AB is a horizontal base. Length of base AB = |7 - 1| = 6 units. The height of the triangle from C(4, 8) to the line containing AB (y=2) is the perpendicular distance, which is the absolute difference in y-coordinates: |8 - 2| = 6 units. Area of triangle = (1/2) * base * height = (1/2) * 6 * 6 = 18 square units.
Why the other options are wrong
- A. Incorrect. This is base*height (6*6), forgetting to divide by 2.
- B. Incorrect. This might be from calculating (1/2)*4*6 or other incorrect base/height.
- D. Incorrect. This is 6*6, not for a triangle.
Area of Triangle (Coordinate Geometry)
The measure of the two-dimensional space enclosed by a triangle whose vertices are given as coordinates on a plane.
- If two vertices share a common x or y coordinate, that side can be used as the base.
- The height is the perpendicular distance from the third vertex to the line containing the base.
- Can also use the Shoelace Formula for any triangle: A = (1/2) |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|.
Memory trick: Find a 'Flat' base, then measure 'Up' for the height.