Here are a few recommended readings before getting started.
Try your knowledge on these topics.
In certain situations, it is required to find the length of a line segment or a side of a polygon. To find those lengths, it is recommended to calculate the distance between the endpoints of the segment, or between two vertices of a polygon. If those points are plotted on a coordinate plane, the Distance Formula can be used.
This formula is called the Distance Formula.
By following the same procedure, it can be shown that and are also right angles.
Since each angle of is a right angle, the quadrilateral is a rectangle. Therefore, Ramsha is correct. She's on her way toward making some nice frames.
The formula above is called the Midpoint Formula.
The midpoint of both diagonals is the same as their point of intersection. Therefore, the diagonals bisect each other.
Any figure on a coordinate plane — located at any position — can be transformed through a combination of rigid motions to have a vertex at the origin and a consecutive vertex on the axis. By this reasoning, when proving a theorem for a figure with a vertex at the origin and a consecutive vertex on the axis, that theorem is valid for any figure with the same shape and size — located at any position on the coordinate plane.
In the applet below, the polygon can be translated by dragging it. Even more, it can be rotated around the center of rotation. Move the center of rotation around any of the polygon's vertices. Try to place a vertex of the polygon at the origin and a consecutive vertex on the positive
Paulina's best friend is obsessed with triangles. He has requested for Paulina to make him a triangular picture frame. Suppose Paulina can prove the Triangle Midsegment Theorem for the triangle below. She will better understand how to work with triangles and therefore make a better frame!
Start by recalling the Triangle Midsegment Theorem.
Triangle Midsegment Theorem
It can be seen above that both and have the same coordinate This means that is a horizontal segment. Since is on the axis, it can be said that it is also a horizontal segment. Therefore, the midsegment is parallel to Finally, it needs to be proven that is half To do this, the Distance Formula will be used.
Since is half the length of is half the length of By following the same procedure, it can be proven that the other two midsegments of are parallel and half the length of the third side. The Triangle Midsegment Theorem has been proven. Paulina has just leveled up and her friend will get a great frame!
The topics covered in this lesson can now be applied to the challenge. The figure below looks like a parallelogram; without using measuring tools, how can it be proven that the quadrilateral is a parallelogram?
To prove that the sides have the same length, the Distance Formula will be used.
The length of the opposite sides and is This means that and are congruent.
The quadrilateral has been proven to have a pair of congruent parallel sides. Therefore, it is a parallelogram.