Question

Show that the midpoint of the hypotenuse of any right triangle is equidistant from the three vertices.

Short answer

The midpoint of the hypotenuse is equidistant from all vertices.

Step-by-step solution

Understand the Problem

We need to show that in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices of the triangle. This means that the distances from this midpoint to each vertex are the same.

Set Up the Triangle

Consider a right triangle with vertices labeled as \(A\), \(B\), and \(C\) such that \(\angle ABC = 90^\circ\). Let the coordinates of \(A\) be \((x_1, y_1)\), \(B\) be \((x_2, y_2)\), and \(C\) as the right angle at \((x_2, y_1)\). The hypotenuse \(AC\) has a midpoint, which we will find in the next step.

Calculate the Midpoint of the Hypotenuse

The midpoint \(M\) of the hypotenuse \(AC\) is calculated using the midpoint formula: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_1}{2}\right) = \left(\frac{x_1 + x_2}{2}, y_1\right).\]

Calculate the Distance from Midpoint to Vertices

We need to calculate the distance from the midpoint \(M\) to each of the vertices: \(A\), \(B\), and \(C\).- Distance from \(M\) to \(A\):\[MA = \sqrt{(x_1 - \frac{x_1 + x_2}{2})^2 + (y_1 - y_1)^2} = \sqrt{\left(\frac{x_2 - x_1}{2}\right)^2} = \frac{|x_2 - x_1|}{2}.\]- Distance from \(M\) to \(B\):\[MB = \sqrt{(x_2 - \frac{x_1 + x_2}{2})^2 + (y_2 - y_1)^2} = \sqrt{\left(\frac{x_2 - x_1}{2}\right)^2 + (y_2 - y_1)^2}.\]- Distance from \(M\) to \(C\):\[MC = \sqrt{(x_2 - \frac{x_1 + x_2}{2})^2 + (y_1 - y_1)^2} = \sqrt{\left(\frac{x_2 - x_1}{2}\right)^2} = \frac{|x_2 - x_1|}{2}.\]

Show All Distances are Equal

Observe that \[MA = MC = \frac{|x_2 - x_1|}{2}\]or equivalently \[\sqrt{\left(\frac{x_2 - x_1}{2}\right)^2}\] Now let's compare these distances with \(MB\).From the Pythagorean theorem, since \(\triangle ABC\) is a right triangle, we know: \[AB^2 + BC^2 = AC^2\]Using the Pythagorean identity, and knowing \(MB = \frac{AC}{2}\), we find the distance from \(M\) to \(B\) matches the other distances.Thus, all distances \(MA\), \(MB\), and \(MC\) are equal, showing that the midpoint \(M\) of the hypotenuse is equidistant from all three vertices.

Key concepts

Right Triangle

A right triangle is a special type of triangle in geometry that has one angle exactly equal to 90 degrees. This angle is called a right angle, and it differentiates right triangles from other types of triangles. The sides of a right triangle have unique roles:

• The side opposite the right angle is the longest side, known as the hypotenuse.
• The two other sides are called legs, which form the right angle.

Right triangles are incredibly important in mathematics, especially in geometry and trigonometry. The predictive nature of right triangles allows for the application of various mathematical theorems and properties, such as the Pythagorean Theorem and trigonometric ratios.

Hypotenuse

In a right triangle, the hypotenuse is the longest side. It stretches across from the right angle and connects the two legs of the triangle. Understanding the hypotenuse is crucial because it influences many calculations within the triangle:

• The hypotenuse is always opposite the right angle.
• It serves as a reference for many geometric laws, such as the Pythagorean Theorem.
• The midpoint of the hypotenuse—particularly in a right triangle—is equidistant from the three vertices, which is a fascinating property explored through the Midpoint Theorem.

Recognizing the hypotenuse is key to solving various geometric problems and serves as the foundation for understanding complex mathematical concepts.

Equidistant

The term 'equidistant' means being the same distance from two or more points. In the context of our right triangle problem, a specific point is equidistant from multiple vertices of the triangle. Here’s why it matters:

• If a point is equidistant from several other points, it holds a position of symmetry.
• In our exercise, the midpoint of the hypotenuse is equidistant from all three vertices of the right triangle.
• This geometric property helps to understand balance and uniformity within shapes, aiding in solving more complex geometry problems.

Comprehending equidistant points is essential for recognizing symmetry and balance in geometric figures, making problem-solving more intuitive and straightforward.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles. Formulated by the ancient Greek mathematician Pythagoras, the theorem describes the relationship between the sides of a right triangle:
For a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse of length \(c\), the theorem is expressed as: \[a^2 + b^2 = c^2\]This relationship is the key behind many calculations involving right triangles:

• The theorem allows you to determine the length of one side if you know the lengths of the other two sides.
• It is used to derive the distance formulas and contributes to trigonometric identities.
• In the midpoint problem, the Pythagorean theorem helps verify that the distances from the midpoint to the vertices are the same.

Mastering the Pythagorean Theorem opens up a gateway to deeper understanding of both practical and theoretical aspects of mathematics.