Problem 26 - Algebra 2

Question

A triangle is rotated \(90^{\circ}\) clockwise about the origin. The coordinates of the vertices are \(F^{\prime}(2,-3), G^{\prime}(-1,-2),\) and \(H^{\prime}(3,-2) .\) What were the coordinates of the triangle in its original position?

Step-by-Step Solution

Answer

Original coordinates: \( F(3, 2), G(2, -1), H(2, 3) \).

Step 1: Understand the Problem

We need to find the original coordinates of a triangle that was rotated 90° clockwise about the origin. We are given the new coordinates after rotation as \( F'(2, -3) \), \( G'(-1, -2) \), and \( H'(3, -2) \).

Step 2: Know the Rotation Transformation

When a point \((x, y)\) is rotated 90° clockwise around the origin, the new coordinates become \((y, -x)\). To find the original coordinates, we reverse this transformation.

Step 3: Reverse the Rotation for Each Vertex

We reverse the transformation by taking each new coordinate \((y', x')\) and reversing it to give \((-x', y')\).

Step 4: Calculate Original Coordinates for F

The new coordinates of \( F' \) are \((2, -3)\). Reverse the transformation to get \(F (-(-3), 2) = F(3, 2)\).

Step 5: Calculate Original Coordinates for G

For \( G'(-1, -2) \), reverse the transformation to get \(G(-(-2), -1) = G(2, -1)\).

Step 6: Calculate Original Coordinates for H

For \( H'(3, -2) \), reverse the transformation to get \(H(-(-2), 3) = H(2, 3)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, often called analytic geometry, blends algebra and geometry using a coordinate system. This system uses pairs of numbers, known as coordinates, to describe the position of points on a plane. The most common coordinate system is the Cartesian coordinate system. In this system, points are defined by a pair of values: one along the horizontal axis (x-axis) and one along the vertical axis (y-axis).

This framework allows us to solve geometric problems algebraically. You can easily calculate distances between points, the midpoint of a line segment, or the area of shapes. Coordinate geometry is essential for understanding shapes like triangles, rectangles, and circles on a plane. When dealing with transformations, such as rotations, this framework lets you see exactly how points move within the plane.

Transformation

A transformation in mathematics is an operation that moves or changes a shape in some way, while preserving its size and structure. Transformations can include translations, reflections, rotations, and scalings. Each type of transformation modifies the coordinates of a shape's points in a specific manner.

In the context of the given problem, we focus on rotations specifically. Rotating a shape involves turning it around a fixed point, which in this case, is the origin of the coordinate system. Understanding transformations, particularly rotations, is crucial because it allows us to visualize and compute how a shape shifts within the coordinate plane.

Triangle Vertices

In any triangle, the vertices refer to the points where the sides intersect. Each triangle has three vertices, often labeled as A, B, and C or, as in the problem, F, G, and H. The position of these vertices in a coordinate plane is defined by their coordinates.

Knowing the vertices of a triangle allows you to determine its shape, size, and position within the plane. When the triangle undergoes transformations such as rotations, the coordinates of the vertices change. Thus, understanding the original and transformed positions of these vertices is fundamental to solving problems involving geometric transformations.

Clockwise Rotation

A clockwise rotation about the origin involves turning each point around the origin in a clockwise direction. It's a common operation in coordinate geometry.

For a 90-degree clockwise rotation, every point \(x, y\) in the plane changes to \(y, -x\). This rule emerges from how angles and positions change in the Cartesian plane. To reverse this transformation, one would swap the x and y coordinates again and change the sign accordingly. This method is key to determining the original position of points after they have been rotated.

Origin

The origin is a fundamental concept in coordinate geometry. It's the point in the Cartesian coordinate system where the x-axis and y-axis intersect, represented by the coordinates \(0, 0\).

The origin serves as the reference point for many geometric transformations. Rotations often occur about the origin, making it essential to understand its role. In this context, the origin acts as the pivot point for the rotational transformation. Hence, knowing the position of the origin helps you correctly apply transformations like the 90-degree clockwise rotation described in the problem.