Question

You need to find 3 things: The coordinates for M, The Length of the Median AM, The coordinates of the Centroid P. Please explain with answers. a) The coordinates for M are (x₁, y₁), The Length of the Median AM is √((x₁ - x)² + (y₁ - y)²), The coordinates of the Centroid P are (x, y). b) The coordinates for M are (x, y), The Length of the Median AM is √((x - x₁)² + (y - y₁)²), The coordinates of the Centroid P are (x₁, y₁). c) The coordinates for M are (x, y), The Length of the Median AM is √((x - x₁)² + (y - y₁)²), The coordinates of the Centroid P are (x, y). d) The coordinates for M are (x₁, y₁), The Length of the Median AM is √((x - x₁)² + (y - y₁)²), The coordinates of the Centroid P are (x₁, y₁).

Answer 1

The correct answer is b). Let me explain:

The centroid (P) of a triangle is calculated using the formula:

P = 1/3 * (x₁, y₁)

This is basically the average of the coordinates of the three vertices of the triangle.

The mid-point (M) is calculated by taking the average of the x-coordinates and y-coordinates of the three vertices of the triangle. That is:

M = (x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3

This forms the coordinates for M which is (x, y).

Next, the length of the median (AM) from the vertex A to the mid-point M is calculated using the distance formula:

AM = sqrt((xM - xA)² + (yM - yA)²)

That is, AM = √((x - x₁)² + (y - y₁)²).

So our M is (x, y), AM is √((x - x₁)² + (y - y₁)²), and P is (x₁, y₁). This is stated in option b). Therefore, the correct option is b).