Problem 25 - Algebra 2

Question

A triangle is rotated \(90^{\circ}\) counterclockwise about the origin. The coordinates of the vertices are \(J^{\prime}(-3,-5), K^{\prime}(-2,7),\) and \(L^{\prime}(1,4) .\) What were the coordinates of the triangle in its original position?

Step-by-Step Solution

Answer

Original coordinates: J(5, -3), K(-7, -2), L(-4, 1).

Step 1: Understand Coordinate Rotation

When we rotate a point \((x, y)\) 90 degrees counterclockwise about the origin, the new coordinates will be \((-y, x)\). The reverse operation, finding the original position from the rotated position, requires reversing this transformation.

Step 2: Reverse Rotate J' to Find J

For \(J'(-3, -5)\), apply the reverse rotation process back to the original position by transforming \((-y, x)\) to \((x, y)\): \((-3, -5) \rightarrow (5, -3)\).Therefore, the original coordinate J was \(J(5, -3)\).

Step 3: Reverse Rotate K' to Find K

For \(K'(-2, 7)\), apply the reverse operation:\((-2, 7) \rightarrow (-7, -2)\). Hence, the original coordinate K was \(K(-7,-2)\).

Step 4: Reverse Rotate L' to Find L

For \(L'(1, 4)\), apply the reverse operation:\((1, 4) \rightarrow (-4, 1)\). Thus, the original coordinate L was \(L(-4,1)\).

Key Concepts

rotation transformation

Imagine you have a shape on a piece of graph paper. When we talk about a "rotation transformation," we're moving this shape around a fixed point, typically the origin \( (0, 0) \). It's like turning a clock's hand but in a mathematical plane. The key ideas here include:

• The center of rotation: Often the origin \( (0, 0) \).
• The angle of rotation: The degree measure you will turn the shape.
• Direction of rotation: Can be either clockwise or counterclockwise.

Using coordinates, we can determine where every point on our shape moves after applying the rotation. For example, to rotate a point \( (x, y) \) by \(90^{\circ}\) counterclockwise around the origin, it transforms into \( (-y, x) \). Understanding this transformation is the key to moving shapes precisely in a coordinate plane.

90 degree rotation

A \(90^{\circ}\) rotation is a specific way to reposition a shape on the graph. Think of the point \( (x, y) \) as moving 90 degrees around into the position \( (-y, x) \). This action is standard and widely used because it's simple to calculate and implement.

When you carry out a 90 degree rotation:

• The x-coordinate becomes the negative of the y-coordinate.
• The y-coordinate becomes the x-coordinate.

This can be recalled easily by thinking of moving from the east to north on a compass. Such rotations are foundational, especially as they don't change the shape's size or form, only its orientation.

counterclockwise rotation

In the math world, counterclockwise rotation moves points in a direction opposite to the clock hands' movement. It's an essential concept in geometry, used for transforming the position of geometric shapes.

When performing a counterclockwise rotation:

• You might visualize turning from "12 o'clock" to "9 o'clock" for a \(90^{\circ}\) rotation.
• The transformation rule for a \(90^{\circ}\) counterclockwise turn is \( (x, y) ightarrow (-y, x) \).

This approach is standard in solving many geometric problems, like the one in our exercise. It's because such rotations tend to follow a positive angular direction on a typical graph, making calculations straightforward and consistent.

reverse rotation process

At times, you need to understand not just how to rotate a shape but how to "reverse" a rotation. This reverse process allows us to trace back where the shape was before it underwent a rotation.

To construct the reverse rotation:

• You take the rotated coordinates \( (-y, x) \).
• Transform them back to the original position \( (x, y) \).

This is like rewinding a movie to see the scenes from before. It's critical in analyzing problems where the final position is known, but you need to determine initial placements, just as performed in the original step-by-step solution of finding the triangle's initial position. Understanding this reverse process expands your toolkit for tackling a variety of geometric transformations.