Question

If the coordinates of the mid points of the sides of a triangle are \((1,2),(0,-1)\) and \((2,-1)\). Find the coordinates of its vertices : (a) \((1,-4),(3,2),(-1,2)(b)(1,2),(2,3),(3,-4)\) (c) \((3,4),(5,2),(1,2)\) (d) none of these

Short answer

Answer: The coordinates of the vertices of the triangle are (1, 2), (3, -4), and (-1, 2).

Step-by-step solution

Assign Midpoints and Unknown Vertices

Assign the given midpoints as M1\((1, 2)\), M2\((0, -1)\), and M3\((2, -1)\). Assign the unknown vertices as A\((x_a, y_a)\), B\((x_b, y_b)\), and C\((x_c, y_c)\).

Identify Midpoint Relationships

Identify the relationships between the midpoints and the vertices: M1 is the midpoint of the side BC, M2 is the midpoint of the side AC, and M3 is the midpoint of the side AB.

Apply the Midpoint Formula to Find Coordinates

Using the midpoint formula, set up equations to find the coordinates of the vertices. For M1 (the midpoint of BC): $$\frac{x_b + x_c}{2} = 1, \frac{y_b + y_c}{2} = 2$$ For M2 (the midpoint of AC): $$\frac{x_a + x_c}{2} = 0, \frac{y_a + y_c}{2} = -1$$ For M3 (the midpoint of AB): $$\frac{x_a + x_b}{2} = 2, \frac{y_a + y_b}{2} = -1$$

Solve the Equations for Coordinates

The given midpoint coordinates can be used to set up 6 linear equations and solve them to find the coordinates of the vertices. Following are the equations:\begin{cases} x_b + x_c = 2 \\ y_b + y_c = 4 \\ x_a + x_c = 0 \\ y_a + y_c = -2 \\ x_a + x_b = 4 \\ y_a + y_b = -2 \end{cases}. Solve these equations to get: $$x_a = 1, y_a = 2$$ $$x_b = 3, y_b = -4$$ $$x_c = -1, y_c = 2$$ The vertices of the triangle are \((1, 2)\), \((3, -4)\), and \((-1, 2)\), which corresponds to option (b).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures using the coordinate plane. This powerful tool allows us to solve geometry problems by transforming them into algebraic equations. In the context of the given exercise, coordinate geometry is used to determine the vertices of a triangle given the midpoints of its sides.

The coordinate plane is a two-dimensional surface with a horizontal line (the x-axis) and a vertical line (the y-axis) intersecting at a point called the origin, marked as (0, 0). Points on this plane are represented by pairs of numbers, known as coordinates, that describe their position relative to the two axes.

For a triangle, the vertices are the points where the sides of the triangle meet. Using the midpoints of the triangle's sides, we can apply formulas from coordinate geometry, such as the midpoint formula, to find the coordinates of these vertices. This approach showcases how algebra and geometry can interplay to solve problems effectively.

Triangle Vertices

A triangle is one of the basic shapes in geometry consisting of three edges and three vertices. The vertices are the corners or the points where its sides meet. When we want to determine the vertices of a triangle in a coordinate plane, we can use known points such as the midpoints of the triangle's sides. The midpoint of a side of a triangle is the point that divides the line segment joining two vertices into two equal parts.

In the given problem, the coordinates of the midpoints are known, and we need to find the vertices' coordinates. Each midpoint essentially provides a piece of the puzzle, and by using the midpoint formula, which is \[\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\], we can create equations that reveal the coordinates of each vertex. Solving these equations algebraically, while understanding the geometric principles at work, leads us to the triangle's vertices. This process illustrates the importance of both visualizing the problem and employing algebraic techniques.

Linear Equations

Linear equations are fundamental in algebra and are used to represent a straight line on the coordinate plane. These equations typically take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, in the context of our exercise, we deal with a system of linear equations that represent the relationships between the coordinates of triangle vertices based on their midpoints.

The midpoint formula generates linear equations without a slope-intercept form because it involves equalities of coordinates directly. In our problem, these equations look like this: \[x_a + x_c = 0\], \[y_a + y_c = -2\], and so on. Solving a system of linear equations requires us to find the values of the unknown variables that satisfy all equations simultaneously.

System of Equations Approach

We employ methods such as substitution or elimination to find the solution to these systems. For example, we could isolate one of the variables in one equation and substitute it into another to eliminate that variable, eventually finding the values of each vertex's coordinates. The solution to these linear equations reveals the precise location of the triangle's vertices on the coordinate plane.