Problem 36 - Algebra 2

Question

GYMNASTICS The drawing at the right shows four positions of a man performing the giant swing in the high bar event. Suppose this drawing is placed on a coordinate grid with the hand grips at \(H(0,0)\) and the toe of the figure in the upper right corner at \(T(7,8) .\) Find the coordinates of the toes of the other three figures, if each successive figure has been rotated \(90^{\circ}\) counterclockwise about the origin.

Step-by-Step Solution

Answer

Coordinates after rotations: 1st: (-8, 7); 2nd: (-7, -8); 3rd: (8, -7).

Step 1: Understanding the Problem

We need to find the coordinates of the toes of three figures that have been rotated 90 degrees counterclockwise about the origin. The starting position is given as point \(T(7,8)\).

Step 2: Applying Rotation Formula

To rotate a point \((x, y)\) 90 degrees counterclockwise about the origin, use the formula \(T'(x, y) = (-y, x)\).

Step 3: First Rotation

Apply the rotation formula to the first position \(T(7,8)\): \[ T'(7, 8) = (-8, 7) \]This is the position of the toes after the first 90-degree rotation.

Step 4: Second Rotation

Apply the rotation formula to the point \(T'(-8, 7)\): \[ T''(-8, 7) = (-7, -8) \]This is the position of the toes after the second 90-degree rotation.

Step 5: Third Rotation

Apply the rotation formula to the point \(T''(-7, -8)\): \[ T'''(-7, -8) = (8, -7) \]This is the position of the toes after the third 90-degree rotation.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry. It uses a coordinate grid to visually represent points, lines, and shapes using algebraic equations.
In a coordinate plane, every point can be identified by an ordered pair \((x, y)\). The origin, \((0, 0)\), is the central point from which all other points are measured in terms of their horizontal \(x\) and vertical \(y\) distances.
By placing figures on this grid, we can perform operations like rotation and translation mathematically.

• Coordinates show positions in space.
• Rotation involves changing these positions using specific formulas.

Understanding the underlying principles of coordinate geometry is essential before diving into complex transformations like rotations.

90-Degree Rotation

A 90-degree rotation involves turning a point around the origin without changing its shape or size.
Imagine spinning an object around a fixed point, like the hand grips in our gymnastics problem.
For any point \((x, y)\) on the plane, a 90-degree rotation counterclockwise results in a new position \((-y, x)\).
Key step:

• Swap the x and y coordinates.
• Change the sign of the new x coordinate.

This rotation is widely used for transformations in both mathematical problems and real-world applications like computer graphics and navigation.

Counterclockwise Rotation

Rotating counterclockwise is like turning the hands of a clock backwards.
When performing a counterclockwise rotation, the points on the grid move in a direction opposite to a clock's hands.
This might feel intuitive if you visualize sitting at the center of a clock and seeing the numbers rotate around you.
In mathematics, counterclockwise is often considered a positive direction, especially in a Cartesian coordinate system, which is crucial for understanding exercises like the giant swing gymnastics problem. Here, each figure moves around the origin in this positive, counterclockwise direction for a succession of perfect 90-degree rotations.

Algebraic Transformation

Algebraic transformation refers to changing the position or appearance of figures using algebraic operations.
In our exercise, rotations are a form of algebraic transformation where we systematically change coordinates.
The transformation formula for a 90-degree counterclockwise rotation is: \[ T'(x, y) = (-y, x) \].
Steps involved include:

• Determine the original coordinates of the point.
• Apply the transformation formula.
• Calculate new coordinates.

Each rotation employs the rotation formula to yield a new set of coordinates, showing how algebra successfully predicts geometric movements.