A dilation by a positive number $r$ about a point
$\cl"red"A$ leaves $\cl"red"A$ fixed, and each other point moves $r$ times as far from
$\cl"red"A$, without changing its direction from $\cl"red"A$.
The figure to the left lets you see several points ($\cl"red"B$,
$\cl"red"C$, $\cl"red"D$, and $\cl"red"E$) get dilated about the point $\cl"red"A$. You can
slide the slider to dilate them by various amounts. The red numbers
to the right of each gray ray tell you how far the original points
$\cl"red"B$, $\cl"red"C$, $\cl"red"D$, and $\cl"red"E$ are from $\cl"red"A$. The
blue numbers to the left of each gray ray
tell you how far the dilated points $\cl"blue"{B'}$, $\cl"blue"{C'}$, $\cl"blue"{D'}$, and
$\cl"blue"{E'}$ are from $\cl"red"A$.
The distance from $\cl"red"B$ to $\cl"red"A$ is 5. What is the
distance from $\cl"red"C$ from $\cl"red"A$?
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How far is $\cl"red"D$ from $\cl"red"A$?
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Slide the slider so you are dilating about $\cl"red"A$ by 1.3. The
distance from $\cl"blue"{B'}$ to $\cl"red"A$ is now 6.5. How
far is $\cl"blue"{C'}$ from $\cl"red"A$?
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With this dilation, how far is $\cl"blue"{D'}$ from $\cl"red"A$?
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The line segments $\cl"red"\ov{AB}$ and
$\cl"red"\ov{AC}$ both end at the point $\cl"red"A$ which we are dilating about. This means that
a dilation by $r$ must multiply the lengths of those line segments by $r$. The figure to the
left lets you investigate what happens to the length of $\cl"red"\ov{BC}$ when you dilate it.
As you slide the slider, the original segment $\cl"red"\ov{BC}$ is drawn in
red, while its dilated version $\cl"blue"\ov{B'C'}$ is drawn in
blue.
What is the length of $\cl"red"\ov{BC}$?
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Slide the slider until you are dilating by 1.4. What is the length of
$\cl"blue"\ov{B'C'}$?
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When you dilate about $\cl"red"A$ by 0.7, what is the length of
$\cl"blue"\ov{B'C'}$?
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When you dilate by any positive number $r$, does it look like the
length of $\cl"blue"\ov{B'C'}$ is equal to its original length times $r$?
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In the last two questions, we saw that dilating by $r$ seemed to take straight
lines to straight lines, preserve angle measures, and multiply all distances by $r$. To see if
these facts are always true for dilations, we’ll start by looking at a picture for $r=2$.
The figure $\cl"blue"{AB'C'}$ to the left is made out of four congruent copies
of the triangle $\cl"red"{▵ABC}$. This means that all of the green
angles (with one tick mark) marked on the figure have the same measure as each other, as
do all of the purple angles (with two tick marks), and all of the
orange angles (with three tick marks).
It looks like dilating about $\cl"red"A$ by 2 takes $\cl"red"B$ to $\cl"blue"B'$,
and $\cl"red"C$ to $\cl"blue"C'$. We’ll start by investigating whether this is true.
The sum of the measures of the angles of $\cl"red"{▵ABC}$ is 180˚
(because the sum of the measures of the angles of any triangle is 180˚). Notice that one angle
of $\cl"red"{▵ABC}$ is green, one is
purple, and one is orange. What is the
sum of the measures of any green angle, any
purple angle, and any orange
angle?
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Of the three marked angles at $\cl"red"B$, one is
green, one is purple, and one is
orange. What is the sum of their three measures?
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Because an angle of 180˚ is a straight line, two line segments that
meet at a point will lie on the same line if the angle between them is 180˚. Is the angle
between $\cl"red"\ov{AB}$ and $\cl"blue"\ov{BB'}$ 180˚?
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Is the path drawn from $\cl"red"{A}$ to $\cl"red"{B}$
to $\cl"blue"{B'}$ a straight line segment?
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Dilating about $\cl"red"A$ by 2 moves $\cl"red"B$ to a point 2 times as far from $\cl"red"A$,
and in the same direction from $\cl"red"A$. If the path drawn from $\cl"red"{A}$ to
$\cl"red"{B}$ to $\cl"blue"{B'}$ is straight, then $\cl"blue"{B'}$ is in the same direction from
$\cl"red"A$ as $\cl"red"{B}$. Also, the distance from $\cl"red"{A}$ to $\cl"blue"{B'}$ would be
$c+c=2c$, while the distance from $\cl"red"{A}$ to $\cl"red"{B}$ is $c$.
If you dilate $\cl"red"{B}$ about $\cl"red"{A}$ by 2, do you get
$\cl"blue"{B'}$?
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What is the sum of the measures of the three marked angles at
$\cl"red"C$?
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Is the path drawn from $\cl"red"{A}$ to $\cl"red"{C}$
to $\cl"blue"{C'}$ a straight line segment?
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If you dilate $\cl"red"{C}$ about $\cl"red"{A}$ by 2, do you get
$\cl"blue"{C'}$?
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Similarly, the sum of the measures of the three marked angles at $\cl"blue"D$ is 180˚.
Is the blue path drawn from $\cl"blue"{B'}$
to $\cl"blue"D$ to $\cl"blue"{C'}$ a straight line segment?
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The segment $\ov{AB'}$ has length $2c$ (twice the length of
the segment $\cl"red"\ov{AB}$) and the segment $\ov{AC'}$ has length $2b$ (twice the length of
the segment $\cl"red"\ov{AC}$). What is the length of the segment $\cl"blue"\ov{B'C'}$?
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What color is the angle $∠C'B'A$, or how many tick marks does it
have? Is it green (with one tick mark),
purple (with two tick marks), or
orange (with three tick marks)?
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Does $∠C'B'A$ have the same measure as $∠CBA$?
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What color is $∠AC'B'$, or how many tick marks does it
have? Is it green (with one tick mark),
purple (with two tick marks), or
orange (with three tick marks)?
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Does $∠AC'B'$ have the same measure as $∠ACB$?
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In summary, we’ve seen that dilation by 2 about $\cl"red"A$ takes $\cl"red"B$ to
$\cl"blue"B'$ and $\cl"red"C$ to $\cl"blue"C'$, doubles the length of $\cl"red"\ov{BC}$ to the
length of $\cl"blue"\ov{B'C'}$, and leaves the measures of the angles in the triangle
$\cl"red"{▵ABC}$ unchanged in the triangle $\cl"blue"{▵AB'C'}$. These facts stay true as you
slide the sliders to move $\cl"red"B$ and $\cl"red"C$. It is also true that the line segment
$\cl"red"\ov{BC}$ stays straight when dilated, which means that it is dilated to the straight
segment $\cl"blue"\ov{B'C'}$. In fact:
When you dilate any figure by 2 about some point $A$, all of its line
segments (not just the ones through $A$) stay straight, all of its distances (not just the ones
through $A$) are multiplied by 2, and all of its angles’ measures (not just the ones at $A$) are
unchanged.
In the last question, you saw that dilating a figure by 2 kept all of its line
segments straight, multiplied all of its distances by 2, and kept all of its angles unchanged.
Now we’ll look at what happens if you dilate by 3.
The figure to the left consists of 9 congruent copies of the triangle $\cl"red"{▵ABC}$.
Notice that the point in the middle of the figure has two green, two
purple, and two orange angles
surrounding it.
The total measure of one green, one
purple, and one orange angle is 180˚.
What is the total measure of two angles of each color?
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Because an angle of 360˚ is a complete circle, a point will be
entirely surrounded by non-overlapping angles if the total measure of those angles is 360˚. Is
the point in the middle of the figure entirely surrounded without overlap by the marked angles?
(That is, do the small triangles fit together exactly at that point?)
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Are the paths from $\cl"red"A$ to $\cl"blue"{B'}$, from $\cl"red"A$ to
$\cl"blue"{C'}$, and from $\cl"blue"{B'}$ to $\cl"blue"{C'}$ straight? (That is, do all the
angles at the intermediate points add up to 180˚?)
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If you dilate $\cl"red"B$ and $\cl"red"C$ about $\cl"red"A$ by 3, do
you get $\cl"blue"{B'}$ and $\cl"blue"{C'}$?
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The segment $\ov{AB'}$ has length $3c$ (three times the length of
the segment $\cl"red"\ov{AB}$) and the segment $\ov{AC'}$ has length $3b$ (three times the
length of the segment $\cl"red"\ov{AC}$). What is the length of the segment
$\cl"blue"\ov{B'C'}$?
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Do $∠C'B'A$ and
$∠AC'B'$ have the same measures as
$∠CBA$ and $∠ACB$?
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In summary, we’ve seen that dilation by 3 about $\cl"red"A$ takes $\cl"red"B$ to
$\cl"blue"B'$ and $\cl"red"C$ to $\cl"blue"C'$, triples the length of $\cl"red"\ov{BC}$ to the
length of $\cl"blue"\ov{B'C'}$, and leaves the measures of the angles in the triangle
$\cl"red"{▵ABC}$ unchanged in the triangle $\cl"blue"{▵AB'C'}$. These facts stay true as you
slide the sliders to move $\cl"red"B$ and $\cl"red"C$. It is also true that the line segment
$\cl"red"\ov{BC}$ stays straight when dilated, which means that it is dilated to the straight
segment $\cl"blue"\ov{B'C'}$. In fact:
When you dilate any figure by 3 about some point $A$, all of its line
segments (not just the ones through $A$) stay straight, all of its distances (not just the ones
through $A$) are multiplied by 3, and all of its angles’ measures (not just the ones at $A$) are
unchanged.
A red triangle $▵ABC$ is drawn to the left. As you
slide the slider to values of $r$ other than 1, extra congruent copies of
$\cl"red"{▵ABC}$ are also drawn, to make a figure $\cl"blue"{AB'C'}$ with sides
$r$ times as long as the sides of $\cl"red"{▵ABC}$.
When you slide $r$ to 3 or more, are the points in the middle of the
figure entirely surrounded without overlap by the marked angles? (That is, do the small
triangles fit together exactly at those points?)
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Are the paths from $\cl"red"A$ to $\cl"blue"{B'}$, from $\cl"red"A$ to
$\cl"blue"{C'}$, and from $\cl"blue"{B'}$ to $\cl"blue"{C'}$ straight? (That is, do all the
angles at the intermediate points add up to 180˚?)
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If you dilate $\cl"red"B$ and $\cl"red"C$ about $\cl"red"A$ by
$r$, do you get $\cl"blue"{B'}$ and $\cl"blue"{C'}$?
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The segment $\ov{AB'}$ has length $rc$ ($r$ times the length of the
segment $\cl"red"\ov{AB}$) and the segment $\ov{AC'}$ has length $rb$ ($r$ times the length of
the segment $\cl"red"\ov{AC}$). What is the length of the segment $\cl"blue"\ov{B'C'}$?
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Do $∠C'B'A$ and
$∠AC'B'$ have the same measures as
$∠CBA$ and $∠ACB$?
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That is, all of the properties we’ve observed about dilations by 2 and 3 are also true of
dilations by larger integers. In fact, they are also true of dilations
by any positive number (integer or not):
When you dilate any figure by $r$ about some point $A$, all of its line
segments (not just the ones through $A$) stay straight, all of its distances (not just the ones
through $A$) are multiplied by $r$, and all of its angles’ measures (not just the ones at $A$)
are unchanged.
Dilations and parallelism
A red line $BC$ is drawn to the left. Slide the
slider to dilate it about the point $\cl"red"A$, producing the blue line
$B'C'$.
The blue marked angle $∠AB'C'$ is the
dilation of the red marked angle $∠ABC$. Are the measures of the two
marked angles equal or unequal?
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The red line $BC$ and the blue line $B'C'$
can be seen as two lines intersected by a transversal, the line
$\cl"gray"{BB'}$. Then the red angle $∠ABC$ and the
blue angle $∠AB'C'$ are corresponding
angles.
Are these corresponding angles equal or unequal in
measure?
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Remember that when corresponding angles are equal in measure, the two
lines intersected by the transversal are parallel. Is the red line $BC$
parallel to the blue line $B'C'$?
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In general:
When you dilate a line $BC$ by $r$ about some point $A$ which is not on the
line, the result is a line parallel to $BC$.