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Translations

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Translations

Study Guide

Key Definition

A translation is a transformation where a figure is moved from one location to another without changing its size, shape, or orientation. For a point $P(x, y)$, translating $a$ units right and $b$ units up results in $P'(x + a, y + b)$.

Important Notes

Translations do not alter the size or shape of figures.
The orientation of the figure remains the same after translation.
In the coordinate plane, translations can be represented by vector additions.
To move left and down, subtract the respective units from the coordinates.
Translations are a type of rigid motion.

Mathematical Notation

$\triangle$ represents a triangle

$\angle$ represents an angle

$+$ represents addition

$-$ represents subtraction

$P'(x + a, y + b)$ is the translated point

Remember to use proper notation when solving problems

Why It Works

Translations are a fundamental type of transformation because they allow figures to be repositioned without alteration to their properties. This is critical in preserving congruence and similarity.

Remember

Use the formula $P'(x + a, y + b)$ to perform translations on the coordinate plane.

Quick Reference

Right and Up Translation:$P'(x + a, y + b)$

Left and Down Translation:$P'(x - a, y - b)$

Understanding Translations

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A translation slides a figure in the coordinate plane without rotation or resizing. Use $P'(x + a, y + b)$ for simple translations.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Translate the point $P(3, 4)$ 2 units right and 5 units up. What is the new point?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If a skateboarder starts at point $Q(-2, 3)$ and moves 4 units left and 7 units down, where does the skateboarder end up?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Translate the vertices of $\triangle ABC$ with $A(1, 2)$, $B(4, 3)$, and $C(6, 1)$ 3 units down and 2 units left.

Challenge Quiz

Single Choice Quiz

Advanced

A rectangle has vertices at $(1, 1)$, $(5, 1)$, $(1, 4)$, and $(5, 4)$. If translated 3 units right and 2 units up, what is the location of the upper right vertex?

Recap

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Review key concepts and takeaways