Problem 14 - Algebra 2

Question

GEOMETRY Triangle MNP has vertices \(M(3,5), N(-2,8),\) and \(P(7,-4) .\) Find the coordinates of the midpoint of each side.

Step-by-Step Solution

Answer

The midpoints are \(\left( \frac{1}{2}, \frac{13}{2} \right)\), \(\left( \frac{5}{2}, 2 \right)\), and \(\left( 5, \frac{1}{2} \right)\).

Step 1: Understanding the Midpoint Formula

The midpoint formula is a way to find the point that is exactly halfway between two given points. For points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the midpoint \(M\) is calculated as follows: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 2: Calculate the Midpoint of MN

Using the coordinates of points \(M(3,5)\) and \(N(-2,8)\), we apply the midpoint formula: \[ M_{MN} = \left( \frac{3 + (-2)}{2}, \frac{5 + 8}{2} \right) = \left( \frac{1}{2}, \frac{13}{2} \right) \] Therefore, the midpoint of MN is \(\left( \frac{1}{2}, \frac{13}{2} \right)\).

Step 3: Calculate the Midpoint of NP

Using the coordinates of points \(N(-2,8)\) and \(P(7,-4)\), we apply the midpoint formula: \[ M_{NP} = \left( \frac{-2 + 7}{2}, \frac{8 + (-4)}{2} \right) = \left( \frac{5}{2}, 2 \right) \] Therefore, the midpoint of NP is \(\left( \frac{5}{2}, 2 \right)\).

Step 4: Calculate the Midpoint of PM

Using the coordinates of points \(P(7, -4)\) and \(M(3, 5)\), we apply the midpoint formula: \[ M_{PM} = \left( \frac{7 + 3}{2}, \frac{-4 + 5}{2} \right) = \left( 5, \frac{1}{2} \right) \] Therefore, the midpoint of PM is \(\left( 5, \frac{1}{2} \right)\).

Key Concepts

Triangle Geometry

Triangles are fundamental shapes in geometry, consisting of three sides, three vertices, and three angles. In triangle geometry, understanding properties such as perimeter, area, and the relations between the sides and angles is crucial.
When dealing with triangles on a coordinate plane, like triangle MNP, we can represent each vertex by a set of coordinates. This allows for precise calculations using algebraic methods rather than just geometric intuition.

• Vertices are the points where two sides of the triangle meet.
• Sides are the line segments connecting each pair of vertices.
• The shape and size of a triangle depend on its vertices' arrangement.

By calculating midpoints, which are the exact centers of each side, we gain insight into the triangle's symmetry and balance.

Coordinate Geometry

Coordinate geometry, or analytical geometry, integrates algebraic principles with geometric concepts. It uses a coordinate system to describe geometric figures numerically.
This branch of mathematics makes it possible to work with shapes on graphs, helping to solve problems involving distance, slope, and intersections more easily.

• Coordinates are pairs of numbers that locate points on a plane. In standard 2D geometry, these are the 'x' and 'y' values.
• Every point on a plane can be described jointly by these coordinates, providing a means to perform calculations and derivations.
• Visualizing triangles on coordinate planes allows for the application of various formulas, like distance and midpoint formulas.

Such an approach simplifies complex geometric problems by translating them into comprehensible algebraic operations.

Calculating Midpoints

Finding the midpoint of a line segment in geometry involves locating the center point between the segment's two endpoints. This concept is particularly useful when analyzing shapes like triangles on a coordinate plane.
To determine a midpoint, use the midpoint formula:

• Identify the coordinates of both endpoints; call them \((x_1, y_1)\) and \((x_2, y_2)\).
• Apply the formula to find the midpoint: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
• The resulting calculation yields a new point which lies exactly midway between the original points on the line.

This methodology is instrumental in dissecting the properties of geometric figures, such as assessing balance and congruency in triangles, like MNP in our exercise.