SLC S23 Week4 || Quadrilaterals

mojociocio (65)member Romaniain WORLD OF XPILAR • 7 days ago

_{Photo taken from Pixabay}

Hello there! You just angled your way into the right place. Today, I’m presenting the homework from @sergeyk's geometry class, so if you're feeling well-rounded you can find it here: SLC S23 Week4 || Quadrilaterals.

Let's start!

Build a parallelogram, demonstrate its elements and properties.

A parallelogram represents a quadrilateral with 2 pairs of parallel sides equal in length and equal opposite angles.

With the parallelogram and the diagonals built and a bit of styling done, this is how it looks:

To demonstrate that the opposite sides are equal, I used the Distance or Length tool just to better illustrate this property. As you can see in the GIF below, no matter how I move the points, the length of opposite sides are equal.

Now let's turn our focus to the angles. For a better view of how the angles behave, I hid the diagonals and the length of the opposite sides.

In the GIF below we can see that the opposite angles have the same value: ∠BCD = ∠BAD and ∠CBA = ∠CDA and the adjacent angles sum to 180° (they are supplementary)

_{Angle behavior}

Let's see the parallelogram with all the elements present and review it's properties:

4 vertices (A, B, C, D)
4 sides: Opposite sides are equal and parallel (AD = BD, AB = CD)
Two pairs of opposite equal angles (∠BCD = ∠BAD and ∠CBA = ∠CDA)
Two diagonals (AC, BD) that bisect each other: AO = OC, BO = OD
Adjacent angles sum to 180° (are supplementary).
The sum of all interior angles is 360°.
Each diagonal splits the parallelogram into two congruent triangles

_{GIF with all elements}

Build a trapezoid, demonstrate its elements and properties.

There are 3 types of trapezoids: Scalene, Isosceles and Right. Let's see how each one is constructed and it's features

Scalene/Arbitrary Trapezoid

When the sides and the angles are not equal, it's called a scalene or arbitrary trapezoid.

Using the steps provided by @sergeyk, I created an arbitrary trapezoid. Let's see the elements and properties of it:

4 vertices (A, B, C, D)
4 sides: 2 parallel (AD,BC) 2 non-parallel (AB,CD)
2 legs (AB, DC)
2 bases (AD, BC)
2 diagonals (AC, BD)
Height (DH)
Midline (FG)
Interior angles sum up to 360°
The segment connecting the mid of the trapezoid legs, called the midline, is parallel to the bases and has the length equal to half the sum of the bases

_{Arbitrary Trapezoid}

Right Trapezoid

Starting with a segment AB, I constructed a perpendicular line through point A and a parallel line to AB. The intersection of these 2 lines is point C:

Next I added a point on the parallel line (D) and the segment DB. Also constructed the segments: AC and CD, added the diagonals, midline and height, angles and some styling:

_{Right Trapezoid}

The elements and properties of right trapezoid:

4 vertices (A, B, C, D)
4 sides: 2 parallel (AD,CD) 2 non-parallel (AC, DB)
Two legs (AC, DB)
One leg is perpendicular to the base (AC)
Two bases (AB, DC)
Two diagonals (AD, BC)
Height (DG and AC)
Two 90° angles (<CAB) and (<ACD)
One pair of parallel sides
The interior angles sum up to 360° with 2 of them being 90°
The segment connecting the mid of the trapezoid legs, called the midline, is parallel to the bases and has the length equal to half the sum of the bases
The height is equal in length to the perpendicular leg

Isosceles Trapezoid

When we have 2 parallel sides and the remaining 2 are equal in length, we have a isosceles trapezoid.


	Starting with a segment and a perpendicular line through each point
	Added a parallel line and the intersection point
	Added point E and segment AE and marked the length
	Using Circle with Center and Radius tool, I created a circle with the center in B and radius equal to AE segment
	The intersection point between the circle and CD segment, point F, is the 4th vertex. Created segment BF
	Clearing the drawing a bit, we are left with this isosceles trapezoid

The elements and properties of a isosceles trapezoid:

4 vertices (A, B, C, D)
4 sides: 2 parallel sides (AB,CD); 2 non-parallel, equal sides (AD, BC)
Equal legs (DA, CB)
Two parallel bases (AB, CD)
Equal base angels (<A = <B) & (<C = <D)
Equal diagonals (AC, BD)
Height perpendicular to the base (DI)
Interior angles sum up to 360°
The segment connecting the mid of the trapezoid legs, called the midline, is parallel to the bases and has the length equal to half the sum of the bases

Let's also take a look at the live representation in the GIF below:

_{Isosceles Trapezoid}

Build a rhombus, demonstrate its elements and properties.

I am fairly certain there are other ways to construct a rhombus in GeoGebra but here is how I did it:

Starting with a segment AB, I added a circle with the center in A and radius AB and added another point on the circle, C, and created the segment AC:

_{Starting point}
Next, I hid the original circle and created 2 more: one with the center in C and radius CA and one with the center in B and radius BA. The 2 circles intersect in a point that will be the 4th vertex of the rhombus:

Added segments BD and CD, constructed the diagonals and did some styling. Also added the angles:

Let's see the elements and properties of a rhombus:

4 vertices: A, B, C, D
4 equal sides: AB, BD, DC, CA
The diagonals bisect each other at 90° (AD, BC)
Opposite angles are equal (<A = <D) & (<B = <C)
Diagonals bisect the angles of the rhombus
The angles sum up to 360°

Build a rectangle, demonstrate its elements and properties.

Creating a rectangle is pretty straight-forward: start with a segment and draw a perpendicular line through each point. Add an arbitrary point on one of the lines and construct the parallel line to the starting segment.

_{Rectangle construction}

The intersection point of the parallel line with the perpendicular line will represent our final vertex.

_Animation

Let's recap the elements and properties of the rectangle:

4 vertices
4 sides: opposite sides are parallel and equal
Interior angles are 90°
Two equal diagonals that bisect each other (not in a 90°)
The angles sum up to 360°

Build a square, demonstrate its elements and properties.

To build the square, I started with a AB segment and created a circle with the center in A and the radius AB, drew 2 perpendicular lines on the segment AB that passes through the 2 points and marked the intersection with the circle:

Next, I added another circle with the center in D and the radius DA, created a secondary radius using the perpendicular line tool and marked the intersection point: