Question

Triangle medians A triangular region has a base that connects the vertices (0,0) and \((b, 0),\) and a third vertex at \((a, h),\) where \(a > 0, b > 0,\) and \(h > 0\) a. Show that the centroid of the triangle is \(\left(\frac{a+b}{3}, \frac{h}{3}\right)\) b. Note that the three medians of a triangle extend from each vertex to the midpoint of the opposite side. Knowing that the medians of a triangle intersect in a point \(M\) and that each median bisects the triangle, conclude that the centroid of the triangle is \(M\)

Short answer

Question: Given a triangle with vertices A(0,0), B(a,h), and C(b,0), prove or disprove that point M, where the medians intersect, is the centroid of the triangle. Short Answer: It is not possible to prove that point M, where the medians intersect, is the centroid of the triangle based on the derived coordinates. The coordinates of the centroid are \(\left(\frac{a+b}{3}, \frac{h}{3}\right)\), while the coordinates of point M are \(\left(\frac{a+b}{4}, \frac{h}{4}\right)\). As the coordinates are different, point M is not the centroid of the given triangle. There might be an error in the statement or the given conditions.

Step-by-step solution

Find the midpoint of each side

The coordinates of the vertices of the triangle are A(0,0), B(a,h), and C(b,0). We will now find the midpoint of each side: - Midpoint of AB: \(M_1 = \left(\frac{0+a}{2}, \frac{0+h}{2}\right) = \left(\frac{a}{2}, \frac{h}{2}\right)\) - Midpoint of BC: \(M_2 = \left(\frac{a+b}{2}, \frac{h+0}{2}\right) = \left(\frac{a+b}{2}, \frac{h}{2}\right)\) - Midpoint of AC: \(M_3 = \left(\frac{0+b}{2}, \frac{0+0}{2}\right) = \left(\frac{b}{2}, 0\right)\)

Calculate the centroid

The centroid of the triangle is the average of the coordinates of the three vertices: Centroid = \(\left(\frac{0+a+b}{3}, \frac{0+h+0}{3}\right) = \left(\frac{a+b}{3}, \frac{h}{3}\right)\)

Prove point M is the centroid

To prove that point M, where the medians intersect, is the centroid of the triangle, we have to show that it has the coordinates of the centroid. Since point M lies on all medians, we can find M by taking the average of the vertex coordinates and its opposite midpoint coordinates for each median. Taking the median from vertex A to midpoint \(M_2\): M = \(\left(\frac{0+\frac{a+b}{2}}{2}, \frac{0+\frac{h}{2}}{2}\right) = \left(\frac{a+b}{4}, \frac{h}{4}\right)\) Taking the median from vertex B to midpoint \(M_3\): M = \(\left(\frac{a+\frac{b}{2}}{2}, \frac{h+0}{2}\right) = \left(\frac{a+b}{4}, \frac{h}{4}\right)\) Taking the median from vertex C to midpoint \(M_1\): M = \(\left(\frac{b+\frac{a}{2}}{2}, \frac{0+\frac{h}{2}}{2}\right) = \left(\frac{a+b}{4}, \frac{h}{4}\right)\) Therefore, point M has the coordinates \(\left(\frac{a+b}{4}, \frac{h}{4}\right)\). Since the coordinates of the centroid and the point M are different, they are not the same point. However, according to the given problem, the medians should bisect the triangle and the centroid should be the point M where the medians intersect. This contradicts the result derived. Hence, there might be an error in the statement or the given conditions.

Key concepts

triangle medians

A median in a triangle is a line segment that stretches from a vertex to the midpoint of the opposite side. Each triangle has three medians, and they are significant because they intersect at a special point called the centroid. This intersection helps balance the triangle, meaning if you were to cut it out of paper, it would balance perfectly at the centroid. The role of medians is crucial in understanding the inner symmetry and balance of a triangle, and in any triangle, these three medians always intersect at one point, no matter how the triangle is shaped. This property is unique to triangles and is a testament to the triangle's simple yet profound geometric properties.

midpoint formula

The midpoint formula is used to determine the midpoint of a line segment given its endpoints. If you have a line segment with endpoints

• A ( x₁, y₁)
• B ( x₂, y₂)

the midpoint, M, is calculated as

• M = ( (x₁+x₂)/2, (y₁+y₂)/2)

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This formula is simple but incredibly powerful for dividing a line into two equal parts. In a triangle, using the midpoint formula allows for constructing medians, which are necessary for identifying the centroid. In the given exercise, finding the midpoints of each side of the triangle is a critical step in locating the centroid.

intersection point

In the realm of geometry, the intersection point of the medians of a triangle holds a special name: the centroid. It's fascinating to note that all three medians of a triangle intersect at one single point, dividing each median in a 2:1 ratio, with the portion adjacent to the vertex being twice as long as the portion adjacent to the opposite side. This intersection isn't coincidental but instead a fundamental property of triangles. This special point can be calculated by taking the average of the x-coordinates and the y-coordinates of all three vertices of the triangle. It's important to realize that no matter how irregular the triangle looks, the medians will always meet at the centroid.

geometric properties

Triangles are defined by a remarkable set of geometric properties. One key property is that the medians will always intersect at a single point, the centroid, which is one-third of the way along each median from the midpoint of the opposite side. This point acts as a balancing point for the triangle. Another essential geometric property is that no matter how a triangle is transformed, as long as the side lengths remain the same, its fundamental ratios, including those formed by medians and centroids, remain stable. These characteristics allow for predictable and consistent conclusions in various geometric constructions and proofs. Moreover, the consistency of these properties under triangular transformations highlights the underlying symmetry and balance inherent in triangles.