Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (2024)

Mastering triangle reflection tests our understanding of transformations and reflections that occur on a rectangular coordinate plane. The triangle is a polygon made up of three points, so we’re observing the reflections of these three points when learning how to reflect triangles on the coordinate system.

Triangle reflection extends our knowledge of reflecting a point on a coordinate system to reflecting three points forming a triangle.

In this article, we’ll show you the process of reflecting a triangle on a coordinate plane. By learning how to reflect these figures over a given line of reflection, we’ll apply our understanding of reflecting points over a coordinate plane. By the end of our discussion, we want you to feel confident when working on reflections of triangles.

What Is a Triangle Reflection?

Triangle reflection is the figure obtained when a triangle is flipped on a coordinate system based on a line of reflection. When studying and working on the reflection of polygons like the triangle, it’s important to know the following terms:

  • Pre-image: The original image (for this discussion, the triangle) that we’re reflecting over a line.
  • Image: The reflected triangle and final version after reflecting the triangle over.

We normally label the image using the pre-image’s points but this time, we add a prime symbol to each of these points’ labels. Let’s take a look at the two triangles plotted on the same $xy$-plane.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (1)

Suppose that the triangle, $ABC$, is the triangle we want to reflect over the $y$-axis or the line, $x=0$. If $ABC$ is the pre-image, then the triangle, $A^{\prime}B^{\prime}C^{\prime}$ is the resulting image after reflecting the triangle.
When working with triangle reflections, the resulting image will retain the shape of the triangle. This means that the lengths and angle measures of these two triangles will be equal.

In triangle reflection, however, the triangle from the pre-image and the image may have different positions. Why don’t we take a look at the points of the triangle, $\Delta ABC$, after being reflected over the $y$-axis?

Pre-Image

Image

\begin{aligned} A= (1, 2)\end{aligned}

\begin{aligned} A^{\prime}= (-1, 2)\end{aligned}

\begin{aligned} B= (4, 4)\end{aligned}

\begin{aligned} B^{\prime}= (-4, 4)\end{aligned}

\begin{aligned} C= (8, 3)\end{aligned}

\begin{aligned} C^{\prime}= (-8, 2)\end{aligned}

We’ve learned that when reflecting points over the $y$-axis, the $x$-coordinate’s sign changes. We extend this concept when reflecting triangles, so the reflection of triangles will depend on the line of reflection as well.

These are the common lines of reflection that you’ll encounter for triangle reflection:

  • The $x$-axis with an equation of $y= 0$
  • The $y$-axis with an equation of $x= 0$
  • The diagonal line with an equation of $y =x$
  • The diagonal line with an equation of $y = -x$

In the next section, we’ll show you how the triangle’s points are affected when the pre-image of the triangle is reflected over these lines. We’ll also show you different examples of reflecting a triangle to help you understand the process better!

How To Reflect a Triangle?

Reflect a triangle by 1) reflecting the three points that form each triangle over the line of reflection and 2) applying the algebraic properties of reflections on each coordinate.

In triangle reflection, the point of the pre-image will have the same distance as that of the image’s point with respect to the line of reflection. This is one way to do this properly.

Now let’s take a look at the triangle $\Delta ABC$. If we want to reflect this over the $x$-axis, the distance of the new triangle’s image must have the same distances as that of points $A$, $B$ and $C$ from the $x$-axis.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (2)

To do so, use the $x$-axis or the line presented by $y = 0$, and measure the distances of $A$, $B$, and $C$.

  • The points $A$ and $C$ are one unit away from the $x$-axis.
  • The point $B$ is 4 units away from the $x$-axis.
  • Reflect the $x$-axis by plotting the image’s points right below the $x$-axis.

Once the reflection’s image is plotted, construct the triangle to show the reflected triangle. Take a look at the image shown below to see how the $\Delta ABC$ is reflected over the $x$-axis.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (3)

We use the same process when reflecting triangles over different lines of reflections. For now, let’s also take a look at how the coordinates change from the pre-image to the image.

Pre-Image

Image

\begin{aligned} A= (1, 1)\end{aligned}

\begin{aligned} A^{\prime}= (1, -1)\end{aligned}

\begin{aligned} B= (4, 4)\end{aligned}

\begin{aligned} B^{\prime}= (4, -4)\end{aligned}

\begin{aligned} C= (5, 1)\end{aligned}

\begin{aligned} C^{\prime}= (5, -1)\end{aligned}

This confirms that when we reflect a triangle over the $x$-axis, we’re simply reflecting the three coordinates by changing the $y$-coordinate’s sign. This means that we can apply the rules of a coordinate reflection to triangle reflection. With this in mind, let’s go ahead and move on to another way of reflecting triangles – by focusing on the vertices’ coordinates.

Here’s a summary of the rules to remember when reflecting the triangles’ coordinates over these four common lines of reflection.

Reflection

Coordinate of the Image

Reflection over the $x$-axis

\begin{aligned} (x, y) \rightarrow (x, -y)\end{aligned}

Reflection over the $y$-axis

\begin{aligned} (x, y) \rightarrow (-x, y)\end{aligned}

Reflection over the line, $y = x$

\begin{aligned} (x, y) \rightarrow (y, x)\end{aligned}

Reflection over the line, $y = -x$

\begin{aligned} (x, y) \rightarrow (-y, -x)\end{aligned}

Reflection over the origin

\begin{aligned} (x, y) \rightarrow (-x, -y)\end{aligned}

The best way to master this topic by heart is through practice. We’ll show you examples and practice questions for you to work on. When you’re ready, head over to the section below!

Example 1

What would the reflection of $\Delta MNO$ look like when reflected over the origin?

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (4)

Solution

To graphically reflect the triangle $\Delta MNO$, first construct a line to guide us in reflecting the triangle over the origin. When reflecting a triangle over the origin, use a line where $(0, 0)$ is the midpoint between $M$ and $M^{\prime}$.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (5)

Now, observe the perpendicular distance of the three vertices from this line.

  • The line passes through point $M$, so it will be passing $M^{\prime}$ through as well.
  • The point, $N$, is roughly $0.5$ unit from the right of the line. This means that point $N^{\prime}$ is approximately $0.5$ unit from the left.
  • Similarly, since $O$ is $4$ units away from the right of the line, $O^{\prime}$ is $4$ units to the left of the line.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (6)

Hence, the result of reflecting $\Delta MNO$ over the origin is the image $\Delta M^{\prime}N^{\prime} O^{\prime}$. If we apply the second method, we can determine the coordinates of the triangle’s image by multiplying the $x$ and $y$-coordinates of each point by $-1$.

Pre-Image

Image

\begin{aligned} A= (2, 4)\end{aligned}

\begin{aligned} A^{\prime}= (-2, -4)\end{aligned}

\begin{aligned} B= (1, 1)\end{aligned}

\begin{aligned} B^{\prime}= (-1, -1)\end{aligned}

\begin{aligned} C= (4, 2)\end{aligned}

\begin{aligned} C^{\prime}= (-4, -2)\end{aligned}

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (7)

This shows that whichever method we use, the result will remain the same. Using the second approach is more efficient for common lines of reflection.

Knowing how to reflect triangles geometrically, however, allow us to work with a wide range of lines of reflection. This means with the two methods in our toolkit, we’ll feel even more confident to work with lines of reflections – both familiar and new.

Practice Question

1. What are the coordinates of the resulting image when $\Delta ABC$ is reflected over the $y$-axis?

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (8)

A. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-2, -5), (2, -1), (4, -4)\}$
B. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(2, 5), (-2, 1), (-4, 4)\}$
C. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-2, 5), (-2, 1), (-4, 4)\}$
D. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(2, 5), (2, 1), (4, 4)\}$

2. What are the coordinates of the resulting image when $\Delta ABC$ is reflected over the $x$-axis?

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (9)

A. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-1, -6), (-3, -1), (4, -2)\}$
B. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-1, 6), (-3, 1), (4, 2)\}$
C. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-1, -6), (3, -1), (-4, -2)\}$
D. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(1, 6), (3, 1), (4, 2)\}$

3. What are the coordinates of the resulting image when $\Delta ABC$ is reflected over the line $y =x$?

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (10)

A. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-6, 2), (-3, -3), (-4, 4)\}$
B. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(6, -2), (3, -3), (4, -4)\}$
C. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(6, 2), (3, -3), (4, 4)\}$
D. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-6, 2), (-3, 3), (-4, -4)\}$

4. What are the coordinates of the resulting image when $\Delta ABC$ is reflected over the line $y = – x$?

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (11)

A. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-5, -4), (-5, -2), (1, -4)\}$
B. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(5, -4), (5, -2), (-1, -4)\}$
C. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(-5, 4), (-5, 2), (1, -4)\}$
D. $\Delta A^{\prime}B^{\prime}C^{\prime} = \{(5, 4), (5, 2), (-1, -4)\}$

Answer Key

1. B
2. A
3. C
4. D

Images/mathematical drawings are created with GeoGebra.

Triangle Reflection - Definition, Techniques, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day (2024)

FAQs

What is the history of triangles in mathematics? ›

The triangle had been discovered centuries earlier in India and China. In the 13th century, Yang Hui (1238–1298) knew of this triangle of numbers. In China, Pascal's triangle is called Yang Hui's triangle. The triangle was known in China in the early 11th century by the mathematician Jia Xian (1010–1070).

What are the 4 ways to represent a mathematical situation? ›

There are four ways for the representation of a function as given below:
  • Algebraically.
  • Numerically.
  • Visually.
  • Verbally.

What is the brief history of mathematics? ›

Prehistoric Africans started using numbers to track time about 20,000 years ago. The Rhind Papyrus (1650 BCE) shows how ancient Egyptians worked out arithmetic and geometry problems in the first math textbook. Babylonian mathematicians were the first known to create a character for zero.

What is the earliest recorded use of mathematics? ›

The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC).

What is the importance of triangle in ancient times? ›

Ancient Egyptians used this group of Pythagorean triples to measure out right angles. They would tie knots in a piece of rope to create 3, 4, and 5 equal spaces. Three people would then hold each corner of the rope and form a right triangle!

What is a triangle in real life math? ›

Sandwiches, traffic signs, fabric hangers, and a billiards rack are all examples of triangles in real life. Ans. Building rafters and curved domes are made of triangles. Some bridges have triangular structures, and the Egyptians built pyramids that are triangular in design.

What are three ways math helped us in ancient times? ›

Applied math developed because of necessity, as a tool to watch the stars and develop calendars, or build architectural marvels.

What is ancient mathematics? ›

The ancient mathematics of the Greeks was a lot like modern mathematics. They're both formal mathematics with axioms, formal definitions, and proofs. In fact, there's a continuous tradition of formal mathematics that goes back about 2400 years. Other ancient mathematics wasn't formal.

What is a simple definition of mathematics? ›

Mathematics is the science and study of quality, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

What was the first thing invented in maths? ›

c. 3400 BC – Mesopotamia, the Sumerians invent the first numeral system, and a system of weights and measures. c. 3100 BC – Egypt, earliest known decimal system allows indefinite counting by way of introducing new symbols.

What is the oldest example of math? ›

The earliest form of mathematics that we know is counting, as our ancestors worked to keep track of how many of various things they had. The earliest evidence of counting we have is a prehistoric bone on which have been marked some tallies, which sometimes appear to be in groups of five.

When did math really begin? ›

The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry.

Who first invented the triangle? ›

Pascal's Triangle History

Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal.

Where does the triangle originate from? ›

Some scholars believe the triangle to be a direct descendant of the ancient Egyptian sistrum. Others do not go quite so far, referring to the triangle as being "allied" with the sistrum throughout history, but not a direct descendant.

What is the history of triangular numbers? ›

However, triangular numbers actually originate from the Pythagoreans, who developed relationships between geometric shapes and numbers; hence the birth of triangular numbers, square numbers, and pentagonal numbers, etc. Students can be introduced to triangular numbers using multilink cubes.

Who discovered a mathematical formula relating to triangles? ›

Pythagoras was a teacher and a philosopher. Pythagoras found out that for a right angle triangle (with one of the angles being 90o), the square of the hypotenuse is equal to the sum of the squares of the other two sides: a2+b2=c2.

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