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Here are a few recommended readings before getting started with this lesson.
Show Pencilbutton. Then, move the pencil tip over the place to mark. Finally, release the click.
A natural history museum manager hopes to place a replica of a moai statue in a triangular room in such a manner that the statue is equidistant from the walls that will have detailed diagrams and explanations. The location of the statue corresponds to the incenter of the triangle formed by the walls. This placement will allow for a clear walking path.
The manager's son, Kevin, sees the blueprint and is excited to help to locate the incenter. Unfortunately, he does not have a protractor. Can he still locate the incenter on the blueprint? If so, how?When copying a segment using a straightedge and compass, the function of the compass is to measure the length of the segment. With this in mind, a string can be a good substitute for the compass.
Using a straightedge and a string, it is possible to construct a copy of a segment.
The following three steps can be used to draw a segment with the same length as $AB.$
With the straightedge, draw a new segment $A_{′}C$ such that it is longer than $AB.$
As can be seen, $AB$ and $A_{′}B_{′}$ are copies of each other.
Using a straightedge and a string, it is possible to construct a copy of an angle.
The following five steps can be used to draw an angle whose measure is equal to the measure of $∠ABC.$
To begin, draw a ray using a straightedge and label its starting point as $E.$
Start by tying a pencil to one end of the string and locating the other end at $B.$ The string must be shorter than the sides of the angle.
Next, holding down the end that is on $B,$ stretch the string. Keeping the string stretched, make an arc that intersects the sides of $∠ABC.$ Let $P$ and $Q$ be these points of intersection.Using the straightedge, draw $EF.$ This ray along with $ED$ form $∠DEF$ whose measure is equal to the measure of $∠ABC.$
Paper folding, also referred to as origami, provides an alternative method for making constructions when tools such as a compass are not on hand.
An angle can be bisected using only a straightedge and folding the paper.
To bisect $∠ABC,$ the following two steps can be done. If available, it is recommended to use tracing paper.
Using the folding paper technique, the challenge presented at the beginning of the lesson can be solved. A natural history museum manager hopes to place a replica of a moai statue in a triangular room in such a manner that the statue is equidistant from the walls that will have detailed diagrams and explanations. The location of the statue corresponds to the incenter of the triangle formed by the walls. This placement will allow for a clear walking path.
The manager's son, Kevin, sees the blueprint and wants to help to locate the incenter. Unfortunately, he does not have a protractor. Can he locate the incenter on the blueprint? If so, how?
Can he locate the incenter? Yes.
How? By folding the blueprint, two different angle bisectors can be drawn. The point where the angle bisectors intersect corresponds to the incenter of the triangle.
Graph:
Draw two different angle bisectors by folding the blueprint. The incenter is the point of intersection of the angle bisectors.
Kevin needs to find the incenter of the triangle formed by the room's walls.
Incenter |
The incenter of a triangle is the point of intersection of the triangle's angle bisectors. |
To obtain the point of intersection, Kevin needs to draw at least two angle bisectors to locate the incenter. To do so, begin by labeling the vertices of the triangle.
To draw the angle bisector of $∠A,$ fold the blueprint so that $AB$ lies on $AC.$ Once the blueprint is unfolded, the crease made will represent the angle bisector.Notice that, thanks to the Incenter Theorem, the angle bisector of $∠B$ does not have to be drawn to locate the incenter.
The usual way to draw the perpendicular bisector of a segment involves a straightedge and a compass. However, the same construction can be done without a compass.
Given a segment drawn on a paper, its perpendicular bisector can be drawn only using a straightedge and a pencil.
To draw the perpendicular bisector of $AB,$ the following two steps can be used. If available, tracing paper is recommended.Math teacher by day, the world's greatest futsal coach by night, Coach Tiffaniqua drew the following formation on a board to teach her players where to position themselves when taking a free kick near the court's center.
Ali, ecstatic, asked where he should be in the formation. Coach Tiffaniqua told Ali that he should be positioned the same distance from Diego, Ignacio, and Kevin. Determine where Ali should be positioned on the futsal field.
Use the fact that the circumcenter — the point of intersection of the triangle's perpendicular bisectors — is equidistant from the vertices.
Notice that Diego, Ignacio, and Kevin form a triangle. Thus, Ali should be positioned at a point that is equidistant from the vertices of this triangle.
By definition, the circumcenter of a triangle is equidistant from the vertices. Therefore, to know where Ali should be placed, find the circumcenter of $△KDI.$ To do so, begin by drawing this triangle on a piece of paper.The last construction of this lesson will be to draw a line perpendicular to a given line through a point that is not on the line. Of course, this can be done using a straightedge and compass. However, it is possible to do so without the use of a compass.
A line perpendicular to a given line through a point not on the line can be drawn by using only a straightedge and a pencil.
To draw a line perpendicular to $AB$ through $P$ the following two steps can be used.
Ali has a piece of paper with an acute triangle drawn on it. He wants to draw the orthic triangle corresponding to $△ABC$ whose vertices are the feet of the altitudes of $△ABC.$
However, Ali does not have a compass at hand. Help Ali draw the desired triangle.
Diagram:
To draw the altitude from $A,$ fold the paper so that $A$ is on the crease and the part of $BC$ in one part of the paper falls above the other part of $BC.$
Before finishing the lesson, two interesting facts about the orthic triangle — also called the pedal triangle.