10.6: Formulas for Justifying Translations
Objectives
The lesson objectives for justifying translations are the following formulas:
 MidPoint Formula
 Distance Formula
 Slope Formula
MidPoint Formula
Concept Content
In the first lessons on transformations, you learned about translations, reflections, rotations, and dilations. The last concept in each of these lessons talked about the properties of the transformation type. One additional property is common to all of these transformations and that is midpoint. The midpoint is the average of the two endpoints in a segment. Midpoint has the symbol and the formula:
Midpoints can be used in proving figures are reflections, for example. Look at the equilateral triangle in the diagram below.
In an equilateral triangle there are three lines of symmetry. A line of symmetry is the same as a line of reflection. What is important for a line of symmetry is that each of the halves that result from drawing a line of symmetry is congruent or is the same size and shape. In the section Reflections of Geometric Shapes you learned about the line of symmetry, which is also called the mirror line.
If you were to draw the lines of symmetry in the triangle above, you would draw a line from each vertex to the midpoint on the opposite side.
is the midpoint of is the midpoint of , and is the midpoint of . The lines and are all lines of symmetry or lines of reflection. You can now say the is reflected about the line to form the .
Guidance
Find the midpoints for the digram below in order to draw the lines of reflection (or the lines of symmetry).
As seen in the graph above, a rectangle has two lines of symmetry.
Examples
Example A
In the diagram below, is the midpoint between and . Find the coordinates of .
Example B
Find the coordinates of point on the line knowing that has coordinates (–3, 8) and the midpoint is (12, 1).
Look at the midpoint formula.
For this problem, if you let point have the coordinates and , then you need to find and using the midpoint formula.
Next you need to separate the coordinate formula and the coordinate formula to solve for your unknowns.
Now multiply each of the equations by 2 in order to get rid of the fraction.
Now you can solve for and .
Therefore the point in the line has coordinates (27, –6).
Example C
Find the midpoints for the digram below in order to draw the lines of reflection (or the lines of symmetry).
As seen in the graph above, a square has two lines of symmetry drawn from the midpoints of the opposite sides. A square actually has two more lines of symmetry that are the diagonals of the square.
Vocabulary
 Line of Symmetry
 The line of symmetry (or the line of reflection or the mirror line) is the line drawn so that each of the halves that result from drawing the line is congruent or is the same size and shape.
 MidPoint
 The midpoint is the average of the two endpoints in a segment. Midpoint has the symbol and the formula:
Guided Practice
1. In the diagram below, is the midpoint between and . Find the coordinates of .
2. Find the coordinates of point on the line knowing that has coordinates (–2, 5) and the midpoint is (10, 1).
3. A diameter is drawn in the circle as shown in the diagram below. What are the coordinates for the center of the circle, ?
Answers
1.
2. Let point have the coordinates and , then find and using the midpoint formula.
Next you need to separate the coordinate formula and the coordinate formula to solve for your unknowns.
Now multiply each of the equations by 2 in order to get rid of the fraction.
Now you can solve for and .
Therefore the point in the line has coordinates (22, –3).
3.
Summary
In this concept you looked at the midpoint formula. The midpoint formula has the symbol and the formula:
Midpoint is a property of all transformations. It is particularly useful in determining the lines of symmetry or the lines of reflection in figures.
Problem Set
Find the midpoint for each line below given the endpoints:
 Line given and .
 Line given and .
 Line given and .
 Line given and .
 Line given and .
For the following lines, one endpoint is given and then the midpoint. Find the other endpoint.
 Line given and .
 Line given and .
 Line given and .
 Line given and .
 Line given and .
For each of the diagrams below, find the midpoints.
Distance Formula
Concept Content
In this second concept of lesson Formulas for Justifying Translations, you will learn how the distance formula is used in transformations. All of the transformations except dilations have the common property of side lengths being the same. Dynamic geometry software can calculate the distance formula but you can calculate it as well. In order to calculate the distance of side lengths, you use the distance formula. The distance formula has the symbol and is calculated using the formula:
Guidance
Triangle has vertices and . The triangle is reflected about the axis to form triangle . Assuming that ,and prove the two triangles are congruent.
To prove congruence, prove that and .
It is given that and , and the distance formula proved that and . Therefore the two triangles are congruent.
Examples
Example A
Line is translated 5 units to the right and 6 units down to produce line . The diagram below shows the endpoints of lines and . Prove the two lines are congruent.
Example B
Line has been rotated about the origin CCW to produce . The diagram below shows the lines and . Prove the two lines are congruent.
Example C
The diamond has been reflected about the line to produce as shown in the diagram below. Prove the two are congruent.
Since the figures are diamonds, you can conclude that all angles are the same and equal to . So you have to prove that and .
Since all of the side measures are the same and and , the two diamonds are congruent figures.
Vocabulary
 Distance Formula
 The distance formula has the symbol and is calculated using the formula
Guided Practice
1. Line drawn from to has undergone a reflection in the axis to produce Line drawn from to . Draw the preimage and image and prove the two lines are congruent.
2. The triangle below has undergone a rotation of CW about the origin. Given that all of the angles are equal, draw the translated image and prove the two figures are congruent.
3. The polygon below has undergone a translation of 7 units to the left and 1 unit up. Given that all of the angles are equal, draw the translated image and prove the two figures are congruent.
Answers
1.
2.
3.
Summary
In this concept of Chapter Is it a Slide, a Flip, or a Turn?, lesson Formulas for Justifying Translations, you were introduced to the distance formula. The distance formula has the symbol and is calculated using the formula:
The distance formula is used to help justify congruence by proving that the sides of a preimage have the same length as the sides of the transformed image.
Problem Set
Find the distance for each line below given the endpoints:
 Line given and .
 Line given and .
 Line given and .
 Line given and .
 Line given and .
For each of the diagrams below, find the distances to prove congruence knowing the angles are congruent.
Slope Formula
Concept Content
For transformations and rotations, the slope of a line can also be found to determine if the lines or figures are congruent. In this last lesson of the chapter, you will use the slope formula to calculate the slope of lines in preimages that have been translated or rotated. The slope formula has the symbol and can be calculated using the formula:
Lines that are parallel have the same slope. In contrast lines that are perpendicular have negative reciprocal slopes. For transformations, the following table shows how the slope formula can be used to help prove congruence.
Transformation Type  Line type  What happens to the slope? 

Translations  Parallel  Slopes will remain the same 
Rotation of  Perpendicular  Slopes will be negative reciprocals 
Rotation of  Parallel  Slopes will be the same 
Dilations  Parallel  Slopes will be the same 
So you can use slope as another justification for transformations including translations, rotations, and dilations.
Guidance
The figure below shows a dilation of two trapezoids. Show the dilation lines have the same slope.
and have the same slopes.
and have the same slopes.
and have the same slopes.
and have the same slopes.
The lines in Image A that correspond to the dilation lines in Image B have the same slope so are parallel.
Examples
Example A
Find the slope of the following line.
Example B
The figure below shows a rotation of two quadrilaterals CW about the origin. Show the slopes of reflected lines have negative reciprocal slopes as one justification of rotations.
The lines in preimage that correspond to the rotated lines in the final image have negative reciprocal slopes so they are perpendicular.
Example C
The figure below shows a translation of line over 4 units to the right and down 3 units. Show the slopes of the translated lines are the same as one justification of translations.
The lines and for the translation are parallel.
Vocabulary
 Slope
 The slope formula has the symbol and can be calculated using the formula:
Guided Practice
1. A line passes through the points (4, 25) and (10, 40). What is the slope of the line?
2. The figure below shows a translation of line over 3 units to the left and up 6 units. Show the slopes of the translated lines are the same as one justification of translations.
3. The figure below shows a rotation of about the origin of Image A to produce Image B. Show the slopes of the rotated lines are the same as one justification of this type of rotation.
Answers
1.
2.
The lines and for the translation are parallel.
3.
The lines in Image A that correspond to the rotated lines in Image B have the same slope so they are parallel.
Summary
In this last concept of lesson Formulas for Justifying Translations, and the chapter Is It a Slide, a Flip, or a Turn?, you learned that the slope can be used as a justification for translations, rotations, and dilations. With reflections, it depends on the type of reflection and the slope cannot always be used as a justification.
The slope formula has the symbol and can be calculated using the formula:
Lines that are parallel have the same slope. In contrast lines that are perpendicular have negative reciprocal slopes. For transformations, the following table shows how the slope formula can be used to help justify congruence.
Transformation Type  Line type  What happens to the slope? 

Translations  Parallel  Slopes will remain the same 
Rotation of  Perpendicular  Slopes will be negative reciprocals 
Rotation of  Parallel  Slopes will be the same 
Dilations  Parallel  Slopes will be the same 
Therefore you can use slope as another justification for transformations including translations, rotations, and dilations.
Problem Set
Find the slope for each line below given the endpoints:
 Line given and .
 Line given and .
 Line given and .
 Line given and .
 Line given and .
Justify the following transformations of the preimage A to the transformed images in the diagrams below using the slope formula.
Summary
In this sixth lesson of Chapter Is it a Slide, a Flip, or a Turn? you learned some of the formulas for justifying transformations. You learned first about the midpoint formula that is used to help justify transformations. It is a property common to all transformations. The midpoint formula has the symbol and is found using the formula:
You next learned about the distance formula. You can use dynamic geometry software to determine distance but you could also calculate the distance given two points. The distance formula has the symbol and is found using the formula:
Finally, you learned in concept 3 the slope formula. You could use the slope formula to justify the property of parallelism if it exists between two or more figures. Remember that parallel lines have the same slope. The slope formula has the symbol and is found using the formula:
Keep in mind that reflections can result in perpendicular lines not parallel ones. Perpendicular lines have negative reciprocal slopes. You see perpendicular lines all the time, such as the lines on a tennis court. (Tennis courts have parallel lines as well!)
The graph above is a sketch of the lines on a tennis court. The green lines are an example of lines that are parallel. The red lines are an example of lines that are perpendicular. You also learned that the parallel and perpendicular slopes depend on the type of transformation. The table below shows the relationship between slope and transformation type.
Transformation Type  Line type  What happens to the slope? 

Translations  Parallel  Slopes will remain the same 
Rotation of  Perpendicular  Slopes will be negative reciprocals 
Rotation of  Parallel  Slopes will be the same 
Dilations  Parallel  Slopes will be the same 
Midpoint, distance, and slope can be used for justification of transformations.
For chapter Is it a Slide, a Flip, or a Turn? you have worked with the four types of transformations, graphed them, wrote notations, worked with the properties of transformations, and then justified them. The table below is a summary of the concepts learned.
Transformation  Notation  Mapping Rule  Properties 

Translation 
1. distance 2. angle measure 3. parallelism 4. colinearity 5. Midpoint 

Reflection 



Rotation 



Dilation 
1. angle measure 2. parallelism 3. colinearity 4. midpoint 
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Date Created:
May 28, 2014Last Modified:
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