In this assignment, we will explore the medial triangle for different types of triangles.  First we need to define the medial triangle.  The medial triangle is a triangle made up of the medians of the original triangle.  Lets look at an example of a medial triangle.

    We can see from the sketch that the medial triangle EFD is located inside of the original triangle ABC.  We can also show that the medial triangle is similar to the original triangle.  Two similar figures have the same shape but are not the same size.  To prove that the two triangles are similar, we need to show that the angles are congruent and the sides all have equal ratios.  First, let's look at the angles for each triangle.

    As you can see from the measures of the angles of the original triangle and the medial triangle, each angle in the original triangle is congruent to the angle opposite it in the medial triangle.  For example, angle CAB is congruent to angle FDE because they have the same measure.  Now we need to look at the ratios of the corresponding sides.

    AC and ED are corresponding sides because they are opposite one another.  The ratio of the side of the original triangle to the side of the medial triangle is 2 to 1.  Now we must show that the other two sets of corresponding sides also have the same ratio.

    In both sets of corresponding sides, the ratio is also 2 to 1.  Therefore, we can say that the two triangles are similar.  We will show in this assignment different types of medial triangles, and we will also show that in each set of medial triangles they are similar to the original triangle.  To construct a medial triangle given the original triangle using a GSP script, click below.

Medial Triangle Script

    If the original triangle is an equilateral triangle, will the medial triangle be an equilateral triangle as well?  Lets look at an example of an equilateral triangle and its medial triangle.

    Next lets look at the measure of the angles and the sides of the original triangle.  The sides must all be equal and the angles must all be equal for the triangle to be an equilateral triangle.

    As you can see from the measurements, the sides and angles for the original triangle are all equal.  Therefore, it is an equilateral triangle.  Next lets look at the angles and sides of the medial triangle.

    As you can see from these measurements, the medial triangle is also an equilateral triangle.  This is true for any equilateral triangle and its medial triangle.  We can also show that any equilateral triangle, and its medial triangle are similar as well.  We already know that the angles for any equilateral triangle are always equal to 60 °.  Therefore, the angles are always congruent.  Now we need to show that the ratio of the corresponding sides are equal.

    As you can see, each set of corresponding sides has the same ratio of 2 to 1.  Therefore, we can make the conjecture that the triangles will always be similar when you have an equilateral triangle and its medial triangle.  To look at a GSP sketch of the equilateral triangle with the measurements of the sides shown, click here.  You can move around the vertices of the original triangle and the triangles will remain similar.

Equilateral Triangle Sketch
 
 

    Next we will look at an example of an isosceles triangle and its medial triangle. We will see in this example if the two triangles are also similar, and if the medial triangle is also an isosceles triangle.

    First lets see if the medial triangle is an isosceles triangle like the original.  An isosceles triangle has two equal sides.

    From the measurements, we can see that the medial triangle is an isosceles triangle since two of the sides are equal.  Now lets see if the two triangles are similar by first measuring the corresponding angles to see if they are congruent.

    As you can see from the measurements, the corresponding angles are congruent because their measures are equal.  Next we need to see if the corresponding sides have equal ratios.

    The ratios are all 2 to 1 for the original triangle to the medial triangle.  Therefore, we can make the conjecture that for all isosceles triangles the medial triangle will also be an isosceles triangle, and the two triangles will be similar.  To look at a GSP sketch of the isosceles triangle, click here.  You can also change the size of the triangle by moving the vertices and the two triangles will both remain isosceles and similar.

Isosceles triangle and its medial triangle.
 
 

    In this set of sketches, we will look at the right triangle and its medial triangle.  We will see if a right triangle's medial triangle is also a right triangle and if the two triangles are similar.  First lets look at a sketch of a right triangle and its medial triangle.

    First lets check to see if the medial triangle is a right triangle by measuring angle FED.

    Since the angle is equal to 90º, the medial triangle is also a right triangle like the original triangle.  Next lets see if the two triangles are similar by seeing if all of the corresponding angles are congruent. We already know that angle ABC is congruent to angle FED since they are both right angles.  Therefore, we need to check the other two pairs of corresponding angles.

    From the measurements, we know that all of the angles are congruent since the measures are equal.  Next lets see if the corresponding sides have equal ratios.

    Each pair of corresponding ratios are equal to 2 to 1.  Therefore, we can say that the two triangles are similar.  Next we will look at an example where we started with the medial triangle and used it to draw the original triangle.  To look at a GSP script to draw an original triangle when given the medial triangle, click here.

Medial Triangle to Original Triangle

Lets say that our medial triangle is a right triangle.

    Next we will draw our original triangle using the medial triangle.

    Next lets see if the original triangle is also a right triangle by measuring angle FED.

    From the measure of angle FED, we can see that the original triangle is also a right triangle.  Next we will see if the two triangles are similar.  First we need to check the other two sets of corresponding angles to see if they are congruent.

    We can see that each set of corresponding angles are congruent.  Next we need to see if the corresponding sides have equal ratios.

    We can see from the calculations that each ratio of the corresponding sides are equal to 2 to 1.  Therefore, we can see from the two right angle sketches that any right triangle will generate a right triangle as its medial triangle.  We can also see that the two triangles will always be similar.

    We have seen that in each example the original triangle and its medial triangle will always be similar.  Therefore, any equilateral triangle will generate an equilateral triangle as its medial triangle.  Any isosceles triangle will generate an isosceles triangle as its medial triangle.  Any right triangle will generate a right triangle as its medial triangle.  We have also seen that the ratio of the corresponding sides of the original triangle to the corresponding sides of the medial triangle will always be 2 to 1.  Each set of corresponding sides and angles in each triangle are opposite each other.
 
 

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