Question

Statement 1 \(\mathrm{ABC}\), is a right angled triangle right angled at \(\mathrm{A}\). Also, \(\mathrm{AB}=\sqrt{5}\) and \(\mathrm{AC}=1\). Then, the length of the median of the triangle through A is \(\sqrt{\frac{3}{2}}\) and Statement 2 Medians of a triangle are concurrent and the point of concurrence is the centroid.

Short answer

Based on the given information, Statement 1 is False because the length of the median through point A is not \(\sqrt{\frac{3}{2}}\), but instead \(\sqrt{6}\). Statement 2 is True, as it is a well-known property that the medians of a triangle are concurrent and their point of concurrence is the centroid.

Step-by-step solution

Find the length of side BC

Using the Pythagorean theorem, we know that the sum of the squares of the two shorter sides is equal to the square of the longest side. Therefore, since the triangle is right-angled at A, we can write: \(AB^2 + AC^2 = BC^2\) Now plug in the given values for AB and AC: \(\sqrt{5}^2 + 1^2 = BC^2\) \(5 + 1 = BC^2\) \(BC^2 = 6\) \(BC = \sqrt{6}\)

Find the midpoint of side BC

Let D be the midpoint of side BC. Since D is the midpoint, BD is equal to half the length of BC. Therefore: \(BD = \frac{1}{2}BC = \frac{1}{2}\sqrt{6}\)

Determine the length of median AD

Now we want to find the length of median AD. In a right-angled triangle, the median to the hypotenuse is half the length of the hypotenuse. Hence: \(AD = \frac{1}{2}AB = \frac{1}{2}\sqrt{5}\)

Apply the formula for the median length

We now compute the length of median AM (where M is the midpoint of BC) using the following formula, where a, b, and c represent the lengths of the three sides of the triangle: \(AM^2 = 2(a^2 + b^2) - c^2\) In our case, a = AC, b = AB, and c = BC. We plug in the values and get: \(AM^2 = 2(1^2 + \sqrt{5}^2) - \sqrt{6}^2\) \(AM^2 = 2(1 + 5) - 6\) \(AM^2 = 12 - 6\) \(AM^2 = 6\) \(AM = \sqrt{6}\) Since AM is not equal to the given length of \(\sqrt{\frac{3}{2}}\), Statement 1 is False. Next, let's address Statement 2:

Verification of Statement 2

Statement 2 states that the medians of a triangle are concurrent and the point of concurrence is the centroid. This is a well-known property of medians that can be proven using vectors or analytic geometry, but this statement is True in general.

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, especially in the context of right-angled triangles. It describes the relationship between the sides of a right triangle. Specifically, it states:

• The sum of the squares of the two shorter sides (often referred to as the legs) is equal to the square of the longest side, which is the hypotenuse.

For a triangle with sides \( a \), \( b \), and \( c \), where \( c \) is the hypotenuse, the theorem is expressed as:\[a^2 + b^2 = c^2\]In the context of the original exercise, this theorem helps to determine the length of side BC by considering the other two sides, AB and AC, both given in the problem. Using the known lengths, we applied:\[\sqrt{5}^2 + 1^2 = BC^2\]This calculation showed that \( BC = \sqrt{6} \). The Pythagorean theorem is one of the cornerstones of Euclidean geometry and is powerful in deriving other properties of triangles.

Median of a triangle

A median in a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has three medians, and they have interesting properties. The median is crucial for various calculations and geometric proofs. In a right-angled triangle, the medians have particular implications:

• For the hypotenuse, the median is half the length of the hypotenuse.
• Medians also help in finding the centroid.

In the given exercise, the median through point A (AD) was of particular interest, starting from vertex A to the midpoint of side BC. Knowing that D is the midpoint, we calculated this using:\[AD = \frac{1}{2}BC = \frac{1}{2}\sqrt{6}\]It helps create balance and symmetry in the triangle, which can sometimes help verify other geometric theorems.

Centroid of a triangle

The centroid is a key point of interest in triangles. It is the point where the three medians of the triangle intersect, and it has some distinct properties:

• It divides each median into two segments, where one segment is double the length of the other.
• The centroid is the 'center of mass' of the triangle and balances the triangle.

Though not directly needed to solve the first statement of the exercise, understanding centroids is crucial for the second statement, as they provide evidence that the medians meet in a single point. In a coordinate plane, you could find the centroid by averaging the coordinates of the vertices. Generally, the centroid serves as an anchor point for analyzing structures within a triangle.

Right-angled triangle properties

Right-angled triangles have several key properties that help solve complex problems:

• One of the angles is exactly 90 degrees, making it a distinctive form of triangle.
• The side opposite the right angle is the longest, known as the hypotenuse.
• Trigonometric functions and Pythagorean theorem are directly associated.

In the given triangle, ABC, the right angle at A provides a simple setup for applying the Pythagorean theorem. Additional properties include using sine, cosine, and tangent ratios to determine unknown sides or angles, when more information is provided. They are widely applicable in fields such as physics, engineering, and various real-world problem-solving scenarios.