6. Take any triangle. Construct a triangle connecting the three midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and one-fourth of its area. Construct G, H, ,C and the I for this new triangle. Compare to G,H, C, and I in the original triangle
Below is a triangle constructed using Geometric Sketchpad. The triangle is represented by points A,B, and C and their corresponding line segments. A line segment is a closed interval corresponding to a finite portion of an infinite line. Line segments are generally labeled with two letters corresponding to their endpoints, say A and B, and then written AB. The length of the line segment is indicated with an overbar. Line segment AB is in red, line segment BC is in yellow, and line segment CA is in blue.
Next, I constructed the midpoints of the three sides of the triangle. The midpoint is the point on a line segment dividing it into two segments of equal length. Click here if you would like to learn more about midpoints.
See the graph below for the for the following midpoints. The point F is the midpoint of AC, E is the midpoint of BC, and D is the midpoint of AB.
Below, the points D, E, and F are the midpoints of their corresponding line segments. Connecting these three midpoints forms a triangle which is shown below as the triangle DEF. This triangle is known as the MEDIAL triangle. It is considered similar to the original triangle which is triangle ABC. It is considered similar to the original triangle because the MEDIAL triangle is one-fourth of the original triangle's area. In addition, the side lengths of the MEDIAL triangle are one-half the length of their corresponding similar side of the original triangle.
Click here to take a more in-depth look at a MEDIAL triangle from Math World's website. Let's do some calculations to see if this is true!
We want to explore the two triangles to see if, in fact, the area of the MEDIAL triange is one-fourth the area of the original triangle's area. We also want to see if the length of the similar sides of the MEDIAL triangle are one-half the length of the original triangle. Let's go!
The formula for the Area of a triangle is A=1/2 (base)(heighth). Click here to learn more about calculating the area of a triangle. For the area of the original triangle ABC, I took the length of the base of BC which was (from above) 9.57 cm. Then after constructing a perpendicular line from point A to the base BC at the point G, I found this distance to be 6.82 cm.
Plugging these figures into the Area formula gives us A = 1/2(9.57 cm)(6.82 cm)=32.6337 cm. This is the Area of the original triangle ABC.
Now, let's compute the area of the MEDIAL triangle to see if it is one-fourth the area of the original triangle. For the MEDIAL triangle, I took the length of the base of DF which was 4.78 cm. Then I constructed a perpendicular line from the point E to the base of DF at the point H and found this distance to be 3.41 cm from above.
Plugging these figures into the Area formula gives us A = 1/2(4.78 cm)(3.41 cm) = 8.1499. This is the Area of the Medial triangle DEF.
So, is the Area of the Medial triangle one-fourth the area of the original triangle?
Is (1/4)(32.6337) = (8.1499)??
Yes, so the first part of our exploration proves true. The Area of the Medial triangle is one-fourth the area of the Original triangle.
Next, we want to compare the side lengths of the original and medial triangles. We want to explore to see if the sides of the medial triangle are one-half the length of the corresponding similar sides of the original triangle.
At a first glance, I thought that the sides CA and DE, BA and EF, and BC and DF were all similar sides. That is why they are color coordinated below as such. Each pair of the similar sides appear to be parallel with one another. After comparing the side lengths, we will be able to determine if, in fact, the sides are similar.
It does appear that the sides I chose were similar sides. So, we have compared the Areas and the similar side lengths of the two triangles. Both explorations show that the Triangle DEF is a Median triangle because it has the necessary characteristics.
For a further investigation, let's compare the centroid (G), orthocenter (H), circumcenter (C), and incenter (I) of the two triangles. Click here to learn more about these constructions and how to sketch them.
The centroid (G) of a triangle is the common intersection of the three medians. Below is the construction of the centroid for the Original triangle and the Medial triangle. It is th epoint in the very center of both triangles. As you can see, the point of the centroid appears to be the same for both triangles. This is an interesting observation.
The orthocenter (H) is the intersection of the three altitudes of a triangle. The altitude of a triangle is just simply its height (the perpendicular length from a point to the opposite segment of a triangle). Below is the orthocenter of the original triangle ABC.
Next, I wanted to find the Orthocenter of the Medial Triangle as well. Below is the construction of the graph with Orthocenters for both triangles.
Since it is hard to see the orthocenters in the above graph, I hid the perpendicular lines and reconstructed the graph below for clarity.
The O-Medial represents the Orthocenter of the Medial Triangle and the O-Original represents the Orthocenter of the Original (larger) triangle. Let's take a closer look to see how the orthocenters of the two triangles are related.
I decided to construct the distance from the orthocenter for each triangle to each side of the corresponding triangle. The distances are given below:
Recalling that the sides CA and DE, BA and EF, and BC and DF are similar sides, I compared the distances for the similar sides to their related orthocenters. It seems that the distance from CA to the orthocenter is 3.70 cm which is half of the distance from DE to the orthocenter or 1.85 cm. After further investigation, it appears that this is the case for the other distances as well.
The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by two points, the Circumcenter (C) is on the perpendicular bisector of each side of the triangle. The circumcenter is also the center of the circumcircle or the circumsctribed circle.
Below is the circumcenter of the original triangle ABC. It was was constructed by forming a perpendicular line at each of the line segment's midpoint. The perpendicular lines are shown in green below.
Next, we want to construct the circumcenter of the Medial triangle to compare to the circumcenter of the Original triangle. The lines perpendicular to the midpoint of the Medial triangle are shown below.
As you can see from above, the distances for the related triangles from each point to each circumcenter is the same. This supports the definition of a circumcenter. It appears from above that the distance of the Medial triangle from the circumcenter to each point is half of that of the original triangle. This is because 2.67 cm is one-half of 5.33 cm. This is another interesting observation!
The incenter (I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides. To construct the incenter, you take the angle bisector of the three angles of the triangle. The Incenter of the original triangle ABC is shown below. The purple lines bisect each angle and intersect at the point known as the incenter. The incenter is also the center of the Incircle or the inscribed circle of the triangle.
Next, we want to look at the incenter of the Medial triangle as well and compare the two incenters. See below for the graph.
After comparing the incenters and their distances to the points of the two triangles, I notice the same ratio exists as in the above previous cases. The distance from the medial triangle's incenter to each point is one-half that of the distance from the original triangle's incenter to each similar point.
For example, remember that point A and point E are similar points. The distance from the incenter of the medial triangle to the point E is one-half that of the distance from the incenter of the original triangle to the point A. This is true for the other points as well.
This concludes the exploration. Let's summarize!
In conclusion, if you are given a triangle, you can construct the midpoints of this triangle which just simply divide each segment of the triangle into two equal segments. Then, after connecting these midpoints, a smaller triangle is formed inside the original triangle and is known as the Medial triangle. The characteristics of this Medial triangle are that is has 1/4 area of that of the original triangle and the similar side lengths of it are 1/2 the length of the original triangle. Finally, after constructing the centroid, orthocenter, circumcenter, and incenter of the two triangles, it has been shown that the ratio of comparison of the medial triangle remains at one-half to that of the original triangle. The centroid was the only construction that actually had the same point for both triangles.
This concludes the investigation of the MEDIAL TRIANGLE.