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## Geometry (all content)

### Course: Geometry (all content)>Unit 10

Lesson 8: 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.

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• There should be a quiz or at least an exercise after the last two videos because I could use some practice with this • This literally doesn't make any sense, any guidance? • why does the bisectors being perpendicular have anything to do with the side equality of the quadrilateral? please explain this to me • 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.
• 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)

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?

P.S. Love Khan Maths program - life changing • You explained it very well its just hard to understand because your not working it out in equations and stuff • So if the lines of symmetry on any quadrilateral are perpendicular, is the shape always symmetrical? • Hi there, why 2 lines with slopes of m and -1/m would be perpendicular? Thanks. • 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
• 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? • Why are the diagonals of a kite perpendicular? I thought kites were defined by having 2 pairs of equal adjacent sides. Thanks in advance! • 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. 