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### Course: High school geometry > Unit 2

Lesson 4: Symmetry# Finding a quadrilateral from its symmetries (example 2)

Two of the points that define a certain quadrilateral are (-4,-2) and (0,5). The quadrilateral has a reflective symmetry over the lines y=x/2 and y=-2x+5. Draw and classify the quadrilateral. Created by Sal Khan.

## Want to join the conversation?

- This literally doesn't make any sense, any guidance?(23 votes)
- So basically, we are trying to draw the quadrilateral and classify it on the graph. (Reccomended to get some graphing paper to follow along (if you do follow along))

First, we know that a quadrilateral has four sides and four vertices (the corners of the quadrilateral)

The problem already provides us with two of the vertices- (-4,-2) and (0,5). Lets just call (-4,-2) point A and (0,5) point B. Let’s also call the line y = ½x line C and the line y = -2x+5 line D.

When the problem says that the quadrilateral is left unchanged by the reflection of the two sloped lines given above, it means that when this quadrilateral is reflected across these two lines, the quadrilateral will have the same exact position in the xy coordinate plane.

First, we sketch the points A and B on the plane. We still don’t know the relation where the two points connect in this quadrilateral yet.

We need more information, hence we sketch the lines we will reflect this quadrilateral over line C and line D. Sketching these two lines, we find out that line C intersects the point A and point B intersects line D! This means that when we:

Reflect the points across y=½x (Line C)

Point A stays the same, since it lies on the line that is being reflected (So A lies on A’)

Because Point B does not lie on line C, B’ would be the same distance away from Point B from the reflecting line, but on the opposite side. Additionally, Line BB’ would be perpendicular to the reflection line C. (Notice how B’ also lies on line D like B does.)

Point B’ would be (4,-3).

The question is: is B’ a vertex on the quadrilateral? Well, it met the first condition, but what about the second condition?

Well, when we get the same quadrilateral when reflecting across line D, B’ would have to remain the same. And guess what? Since B’ lies on the reflecting line D, It doesn’t change, meaning B’ is a vertex of our quadrilateral.

Now what about the last remaining vertex? Well, since previously point A was on line C, that means that the last vertex would most likely follow the reflection of A’ over line D, since that is the only rule it needs to follow if A lies on line C.

Applying the same reflective properties we get A’ to be

A’ = (8,4)

Now, we can connect the dots to create a quadrilateral. But the question is: what type of quadrilateral is it?

If you sketched it out or looked at the quadrilateral in the video, it looks like a rhombus. But to check, we use the distance formula on all of the points.

A rhombus must have four equal side lengths.

The distance formula is:

d= √((x_2-x_1)^2+(y_2-y_1)^2) where:

x_1 is the x coordinate of the first coordinate pair

y_1 is the y coordinate of the first coordinate pair

x_2 is the x coordinate of the second coordinate pair

y_2 is the y coordinate of the second coordinate pair

Actually, scratch that. That would be too complicated. However, as long as the diagonals of the opposite vertices intersect at a perpendicular angle, it is a rhombus. It turns out that the diagonals of the opposite vertices are our reflecting lines C and D.

Line C has a formula of y= 1/2x where the slope m=½.

Line D has a formula of y=-2x+5 where the slope m=-2.

We know that in order for the lines to be perpendicular, their slopes must be the negative reciprocals of each other. Since -2 is the negative reciprocal of ½ and vice versa, this quadrilateral is indeed a rhombus!

(If you thought this was too long, rewatch the video again until you understand this. Sal explains it much better and probably quicker in the video, and although you may not get it at first, just keep trying 👍)(2 votes)

- why does the bisectors being perpendicular have anything to do with the side equality of the quadrilateral? please explain this to me(10 votes)
- First off, the property of parallelograms are that opposite sides are parallel and congruent. The property of kites are that diagonals are perpendicular. So the only thing that is both a kite and a parallelogram is a rhombus (since it cannot be a square because the slopes of sides are not perpendicular. We can always check by Pythagorean Theorem:

7^2 + 4^2 = 49+16 = 65 so these sides are √65

8^2 + 1^2 = 64 + 1 = 65 so these sides are also √65

Since all sides are equal, and vertices are not perpendicular, we verify that rhombus is the correct answer.(23 votes)

- Hi there, why 2 lines with slopes of m and -1/m would be perpendicular? Thanks.(7 votes)
- Let the lines

𝑦 = 𝑚₁𝑥 + 𝑏₁ and 𝑦 = 𝑚₂𝑥 + 𝑏₂

be perpendicular to each other.

Shifting the lines vertically doesn't change the fact that they are perpendicular,

so it's actually enough to consider the lines

𝑦 = 𝑚₁𝑥 and 𝑦 = 𝑚₂𝑥

– – –

One point on the first line is 𝑃₁ = (1, 𝑚₁)

One point on the second line is 𝑃₂ = (1, 𝑚₂)

Also, the lines intersect at the origin 𝑃₃ = (0, 0)

– – –

Because the two lines are perpendicular, these three points form a right triangle, with the right angle at 𝑃₃.

According to the Pythagorean theorem we then have

|𝑃₂ − 𝑃₁|² = |𝑃₃ − 𝑃₁|² + |𝑃₃ − 𝑃₂|²

– – –

Using the distance formula, we find

|𝑃₂ − 𝑃₁| = √((1 − 1)² + (𝑚₂ − 𝑚₁)²) = √(𝑚₂ − 𝑚₁)²

⇒ |𝑃₂ − 𝑃₁|² = (𝑚₂ − 𝑚₁)²

|𝑃₃ − 𝑃₁| = √((0 − 1)² + (0 − 𝑚₁)²) = √(1 + 𝑚₁²)

⇒ |𝑃₃ − 𝑃₁|² = 1 + 𝑚₁²

|𝑃₃ − 𝑃₂| = √((0 − 1)² + (0 − 𝑚₂)²) = √(1 + 𝑚₂²)

⇒ |𝑃₃ − 𝑃₂|² = 1 + 𝑚₂²

– – –

So,

(𝑚₂ − 𝑚₁)² = 1 + 𝑚₁² + 1 + 𝑚₂²

⇒ 𝑚₂² − 2𝑚₂𝑚₁ + 𝑚₁² = 2 + 𝑚₁² + 𝑚₂²

⇒ −2𝑚₂𝑚₁ = 2

⇒ 𝑚₂ = −1∕𝑚₁(17 votes)

- so if e=mc@36 shouldent that divide the photosynthis of the quatiratical equasion to 5.37.55 so the equation is 34.55.66=456+66/55====55=66/5=1?(3 votes)
- You lost me at "so if"

I don't mean to be rude, but...

what on earth do you mean?

Could you please put this in simpler terms for everyone?(20 votes)

- There isn't a comprehensive video on rotational symmetry and no space to ask on the practice tests but the test question reads:

One of the points that defines a certain quadrilateral is (1,1). The quadrilateral has rotational symmetry of 90∘ degrees about the point (−4,−4)

And the first answer/hint reads:

Performing the rotation brings the starting point to (−9,1)

How do people know this? Using what formula did they arrive at this answer? Have I missed a video? Intro to rotational symmetry doesn't answer any of these questions and certainly doesn't give sufficient examples.

Also, on the same practice test, they ask if lines dividing a given geometric figure do so in such a way that creates symmetry and some questions you can eyeball but some are very precise and only off by a slight margin and in the answers it just shows arrows - how do they decide the slope of these arrows to ensure symmetry? How should I do this at home? Should I be calculating the slope of each line segment to ensure all slopes are identical? And if so, was this in a video that I missed somewhere?

Please help.

P.S. Love Khan Maths program - life changing(11 votes) - So if the lines of symmetry on any quadrilateral are perpendicular, is the shape always symmetrical?(6 votes)
- If a figure has
*any*line of symmetry, it must be symmetric.(10 votes)

- You explained it very well its just hard to understand because your not working it out in equations and stuff(8 votes)
- There are little to no equations in finding a quadrilateral from its symmetries.(5 votes)

- im so confused(8 votes)
- How did he get the reflection line of y=-2x+5? He got y=0 and x=5(5 votes)
- Ok, I solved this on my own.

y=-2x+5

Choose x = 0 and solve for y. y = -2x + 5. y = (-2) (0)(multiply -2 by 0 and you get 0) + 5 = 5. Now you are left with y = 5

Which gives us x = 0, and y = 5, which is Sal's answer.

Hope this helps!

Calc-U-Later!(7 votes)

- Why are the diagonals of a kite perpendicular? I thought kites were defined by having 2 pairs of equal adjacent sides. Thanks in advance!(4 votes)
- Draw a kite ABCD. (Actually draw it.) Draw the diagonals, and say they intersect at E.

Because it's a kite, AB=BC. So triangle ABC is isosceles, and angle BAE=angle BCE.

Also, because it's a kite, CD=DA, and BD is equal to itself. So triangles ABD and CBD are congruent by SSS.

So their corresponding angles are congruent. So angle ABE=angle CBE.

So by SAS, triangle AEB is congruent to triangle CEB.

So angle AEB is congruent to angle CEB.

But AEB and CEB are also supplementary. So they must be right angles.

So the diagonals of a kite are perpendicular.(6 votes)

## Video transcript

Two of the points that define
a certain quadrilateral are negative 4 comma negative 2. So let's plot that. So that's negative
4 comma negative 2. And 0 comma 5. So that's 0 comma
5 right over there. The quadrilateral
is left unchanged by a reflection over the
line y is equal to x over 2. So what does that
line look like? y is equal to x over 2. I'll do that in the blue. y is equal to x over 2. So when x is equal to 0, y is 0. The y-intercept is 0 here. And the slope is 1/2. Every time x increases by
1, y will increase by 1/2. Or when x increases by
2, y will increase by 1. So x increases by
2, y increases by 1. X increases by 2,
y increases by 1. Or another way to think about
it, y is always 1/2 of x. So when x is 4, y is 2. When x is 6, y is 3. When x is 8, y is 4. So we can connect these. Let me try my best
attempt to draw these in a relatively
straight line. And then I can keep going. When x is negative
2, y is negative 1. When x is negative
4, y is negative 2. So it actually goes through
that point right there. And it just keeps going
with a slope of 1/2. So this line, and I can draw
it a little bit thicker now, now that I've dotted it out. This is the line y
is equal to x over 2. And they also say that
the quadrilateral is left unchanged by reflection
over the line y is equal to
negative 2x plus 5. So the y-intercept here is 5. When x is 0, y is 5. So it actually goes
through that point. And the slope is negative 2. Every time we increase by 1-- or
every time we increase x by 1, we decrease y by 2. So that would go there. We go there. And we keep going at
a slope of negative 2. So it's going to look
something like this. It actually goes
through that point and just keeps going on and on. So this is my best attempt
at drawing that line. So that is y is equal
to negative 2x plus 5. Now let's think about it. Let's see if we can
draw this quadrilateral. So let's first reflect
the quadrilateral, or let's reflect the points
we have over the line y is equal to x over 2. So this is the line y
is equal to x over 2. This magenta point,
the point negative 4, 2 is already on that line. So it's its own reflection,
I guess you could say. It's on the mirror is one
way to think about it. But we can easily reflect
at this line over here. This line, if we were to drop
a perpendicular-- And actually, this line right over here,
y is equal to negative 2x plus 5 is perpendicular
to y is equal to x over 2. How do we know? Well if one line
has a slope of m, then the line
that's perpendicular would be the negative
reciprocal of this. It would be negative 1 over m. So this first line
has a slope of 1/2. Well what's the negative
reciprocal of 1/2? Well the reciprocal
of 1/2 is 2/1. and you make that negative. So it is equal to negative 2. So this slope is a negative
reciprocal of this slope. So these lines are
indeed-- I'm trying to erase that-- perpendicular. So we literally could
drop a perpendicular, literally go along this
line right over here, in our attempt to reflect. And we see that we're
going down 2, over 1 twice. So let's go down 2, over 1,
down 2, over 1, twice again. Let me do that in
that same color. The reflection of this point,
across y is equal to x/2 is this point right over there. So now we have three points
of our quadrilateral. Let's see if we
can get a fourth. So let's go to
the magenta point. The magenta point
we've already seen. It's sitting on top
of y equals x/2, so trying to reflect it
doesn't help us much, but we could try to
reflect it across y is equal to negative 2x plus 5. So once again these lines are
perpendicular to each other. Actually let me mark that off. These lines are perpendicular. So we can drop a perpendicular
and try to find its reflection. So we're going to
the right 2 and up 1. We're doing that once, twice,
three times on the left side. So let's do that once, twice,
three times on the right side. So the reflection
is right there. We essentially want
to go to that line. And however far we
were to the left of it, we want to go that same, that
bottom left direction, which you want to go in the same
direction, to the top right, the same distance to
get the reflection. So there you have it. There is our other point. So now we have the four
points of this quadrilateral. Four points of this
quadrilateral are-- or the four sides, let me actually just
draw the quadrilateral. We have our four points. So this is one side
right over here. This is one side,
right over here. This is another side,
right over here. And you can verify that
these are parallel. How would you verify
that they're parallel? Well they have the same slope. To get from this point to that
point, you have to go over, so your run has to
be 4, and you have to rise 1, 2, 3, 4, 5, 6, 7. So the slope here is 7/4. Rise over run, or change in
y over change in x, is 7/4. And over here,
you go 1, 2, 3, 4. So you run 4 and then you
rise 1, 2, 3, 4, 5, 6, 7. So the slope here is also 7/4. So these two lines are
going to be parallel. And then we could draw
these lines over here. So this one at the top. The one at the top
right over there. And what's its slope? Well, let's see. We go from x equals
0 to x equals 8. So we go down, our change in
y is negative 1 every time we increase x by 8. So this is the slope of--
slope is equal to negative 1/8. And that's the exact same slope
that we have right over here. Negative 1/8. So these two lines
are parallel as well. This line is parallel
to this line as well. So at minimum, we're dealing
with a parallelogram. But let's see if we can
get even more specific. Because this kind of
looks like a rhombus. It looks like a
parallelogram, where all four sides have
the same length. So there's a couple
of ways that you could verify that this
parallelogram is a rhombus. One way is you
could actually find the distance between the points. We know the coordinates, so you
could use the distance formula, which really comes straight
out of the Pythagorean theorem. Or even better, you could
look at the diagonals of this rhombus-- we could
look at the diagonals of this parallelogram. We are trying to figure
out if it's a rhombus. And if the diagonals
are perpendicular, then you're dealing
with a rhombus. And we've already shown
that these diagonals, that this diagonal, this
diagonal, and this diagonal are perpendicular. They intersect at right angles. And so this must be a rhombus.