The Midline Theorem by Myca Pelayo is a part of Part C.
Learning Math: Geometry is 5.The Midline Theorem is part of Dissections and Proof.
We solved Problem B3 by using a triangle to form a parallelogram.note 4
The cut was at the middle of the two segments.The following are some of the questions that we will be able to answer.
There is a distinction between an object and a measurement.The tradition of mathematicians being careful with their words carries over into how we write the names of things.For example:
The segment has endpoints X and Y.The line over the letters tells you that we are talking about a segment, not a line or a ray.
The length of the segment with endpoints X and Y is referred to as XY.The meaning is changed by the lack of segment above the letters.We are talking about the length of that object, not the object itself.
When comparing segments, this distinction continues.We say that there are two segments that are congruent.
The same information is conveyed by both statements, but one is talking about segments and the other about lengths.
There is a distinction between angles and their measures.The angles are congruent.
A segment that is parallel to the base and half as long can be created by cutting along the midline of a triangle.Is that reasonable?
To prove the midline cut works, you need to use geometry facts that you have already encountered.Take some time to think about why these statements are true.(Note 5)
The angles opposite each other when two lines intersect are congruent.
Since the two angles together create a straight angle, we can show why.We can see that m2 is subtracted from each part of the equation.
If the angles between the two sides of the triangles are the same, they are congruent.All three sides of the two triangles have the same length and measure, and all three angles have a similar size and shape.This is known as SAS.
The two sides are the same length.The two sides are the same length.The two angles have the same measure.
The easiest way to be sure that the two triangles are congruent is to draw them together.With a 60 angle between the two segments, draw a segment 2 inches long.Is there more than one way to complete the triangle?You can come up with other cases.
A segment parallel to the base and half as long is created by cutting along the midline of a triangle.
This is the picture to start with.D and E are the same as AC and BC.D, E, and F are on the same line.There is a note 6.
Since we proved the midline theorem, we can call it a theorem.
Being able to make these kinds of proof is useful in the professional world.In order to ensure accurate dimensions, Mr. Ialeggio proves that the window frames he makes for antique homes are indeed rectangles.
The session video has this segment on it approximately 23 minutes and 29 seconds after the logo.
In the picture below, both BC and AD are shown.Explain why the two are the same length.What type of quadrilateral is it?
If you want to connect the four sides in order, draw another quadrilateral.What kind of quadrilateral is it?There is a note 7.
Problem C1-C6 was developed by Educational Development Center, Inc.Glencoe/McGraw-Hill is a trademark.Permission was used.glencoe.com has a math section.
If you are working in a group, you should check to see if anyone uses a cutting method similar to the one presented here.If that's the case, consider using their method instead of the one shown here.
Take a few minutes to think about each fact.Posters that everyone can refer to as they work on the proof may be helpful if you are working in a group.
It can be hard for people to separate this figure from their cutting activities.It might be helpful to create the figure yourself if you are clear about how it was constructed.Start with the triangle.Find midpoints D and E.That segment should be extended to twice its length.We know what AD, BE, E, and DE are; angle DEF is 180; and we made the figure that way.All we know is that.
In the Take It Further section of Session 4, you may have made a hypothesis about this situation.You have the tools to prove it.
The sides are congruent because E is a part of BC.The sides are congruent.Since they are vertical angles, the angles between the congruent pairs of sides are equal.The two triangles are congruent.
The sides of the two triangles are congruent.DC and FB have the same length.DC has the same length as AD, since D is the middle of AC.The length of AD and BF is the same.
The angles DCE and FBE are the same as they are incongruent triangles.The two angles along a transversal to the two lines become alternate interior angles if we extend line segments DC and FB into lines.If the two lines are parallel, we conclude that the alternate interior angles are congruent.We have shown that the segments are related.
By Problems C2 and C3 we know that the segments are parallel.They are both in the same area.Since we have one pair of opposite sides congruent and parallel, we can use the fourth definition in Fact 3 to conclude thatABFD is a parallelogram.
They are both parallel since they are opposite sides of a parallelogram.The points D, E, and F all lie on the same line.DE is also close toAB.
Since they are opposite sides of a parallelogram, they must have the same length.Both DE and EF have the same length.DE must be half as long asDF.DE is half as long as the other two.
The midline of ABC is parallel to AC and half the length of AC.Since HG is a midline of ACD, it is parallel to AC and half its length.Since they are both half the length of AC, EF and HG are equal in length.The EFGH is a parallelogram since it has a pair of congruent and parallel opposite sides.
ABCD was constructed to be a kite, so its diagonals are not straight.The sides of the inside quadrilateral are parallel to one another.There is a special kind of parallelogram called a rectangle.
Problem C6 can be used to prove that the inscribed quadrilateral must always be a parallelogram.The diagram below shows how to make the diagonal BD.The base of the two triangles is the diagonal, and we have FG and EH in the middle.Each midline is parallel to the base.By using diagonal AC, we can say that the two must be parallel to one another.According to definition 1 in Fact 3, no matter how the original quadrilateral is constructed, EFGH must be a parallelogram.
It is not a parallelogram if the sides of the interior quadrilateral coincide.In rare cases, it will be a parallelogram.
Geometric thinking can be explored using rich visualization problems and mathematical language.Geometric construction can be done using intuition and visual tools.Take a look at the basic objects of geometry.
You can learn about the different properties of triangles.Look at the conditions that cause triangle inequality, triangle rigidity, and side–side side congruence.How are these concepts applied to quadrilaterals?Building structures are examples of practical applications of triangles and quadrilaterals.
You can explore the properties of polygons through puzzles and games.Look at mathematical definitions in a more formal way.
To examine the properties of parallel lines and circles, use dynamic geometry software to construct figures with given characteristics, such as segments that are parallel or equal in length.You can find the properties and relationships among geometric figures by looking past formal definitions.
Look at transformations such as translation, reflection, and rotation.These ideas can be used to solve more complex geometric problems.You can use your knowledge of figures to solve problems.The midline theorem can be analyzed and proved.
Continue to look at the idea of proof.One of the most famous theorems in mathematics is the Pythagorean theorem.The distance formula is an application of the Pythagorean theorem.
One of the most important ideas in mathematics is symmetry.Geometric notions of symmetry can be explored by creating designs.Learn about frieze patterns after investigating line symmetry and rotation symmetry.
If you want to progress toward a more formal definition of similarity, you need to examine your intuitive notions of what makes a good copy.Take a look at some applications of similar triangles.
There are various aspects of solid geometry.There are a finite number of platonic solids.Nets and cross–sections can be used to establish relationships between two and three–dimensional geometry.