Question

The sides of a triangle are \(5 x+12 y, 12 x+5 y\) and \(13 x+13 y\) where \(x, y\) are positive numbers. Then the triangle is (a) right-angled (b) obtuse angled (c) acute angled (d) equilateral

Short answer

Answer: Right-angled triangle

Step-by-step solution

Verify the Triangle Inequality Condition

First, let's ensure the triangle given by the sides \(5x + 12y\), \(12x + 5y\), and \(13x + 13y\) is valid by using the Triangle Inequality Condition, which states that for a triangle with sides a, b, and c: a + b > c b + c > a c + a > b For our triangle, let's call side a = \(5x + 12y\), side b = \(12x + 5y\), and side c = \(13x + 13y\). We need to check the following inequalities: a + b > c: \((5x + 12y) + (12x + 5y) > (13x + 13y)\) b + c > a: \((12x + 5y) + (13x + 13y) > (5x + 12y)\) c + a > b: \((13x + 13y) + (5x + 12y) > (12x + 5y)\) The inequalities are satisfied because all the coefficients of x and y are positive. Now that we have verified the given sides can form a triangle, let's determine what type of triangle it is.

Calculate the Square of the Side Lengths

Let's label the sides as follows: \(a^2=(5x + 12y)^2\) \(b^2=(12x + 5y)^2\) \(c^2=(13x + 13y)^2\) Now, calculate the squares of the side lengths: \(a^2=(5x + 12y)^2 = 25x^2 + 120xy + 144y^2\) \(b^2=(12x + 5y)^2 = 144x^2 + 120xy + 25y^2\) \(c^2=(13x + 13y)^2 = 169x^2 + 338xy + 169y^2\)

Determine the Type of Triangle

To determine the type of triangle, we will compare the square of the sides. If either a right triangle, an obtuse-angled triangle, or an acute-angled triangle is formed, only one of these conditions can hold true: a) Right-Angled Triangle: \(a^2 + b^2 = c^2\) b) Obtuse-Angled Triangle: \(a^2 + b^2 < c^2\) c) Acute-Angled Triangle: \(a^2 + b^2 > c^2\) Let's check for each condition: a) \(a^2 + b^2 = (25x^2 + 120xy + 144y^2) + (144x^2 + 120xy + 25y^2) = 169x^2+338xy+169y^2 = c^2\) Since the equality holds, the triangle with sides \(5x + 12y\), \(12x + 5y\), and \(13x + 13y\) is a right-angled triangle. So, the correct answer is (a) right-angled triangle.

Key concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry that ensures the possibility of forming a triangle with given side lengths. According to this theorem, for any triangle with sides denoted as \(a\), \(b\), and \(c\), the following conditions must be met:

• \(a + b > c\)
• \(b + c > a\)
• \(c + a > b\)

These inequalities guarantee that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This helps confirm whether a set of three side lengths can actually form a triangle. In the exercise, we apply this theorem to the sides \(5x + 12y\), \(12x + 5y\), and \(13x + 13y\), and verify that all conditions are satisfied, ensuring these sides can form a valid triangle.

Right-Angled Triangle

A right-angled triangle is a special type of triangle where one of its angles is exactly 90 degrees. This property makes it unique and is crucial when calculating side lengths or applying trigonometry. In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The relation between the sides is defined by the Pythagorean Theorem:

• The square of the hypotenuse = Sum of the squares of the other two sides.

Mathematically, for sides \(a\), \(b\), and hypotenuse \(c\), it is expressed as:
\(a^2 + b^2 = c^2\)
In our exercise, the sides \(5x + 12y\), \(12x + 5y\), and \(13x + 13y\) satisfy this condition. Here, \(13x + 13y\) is the hypotenuse, ensuring the triangle is right-angled.

Calculating Side Lengths

To determine the type of triangle, it is often necessary to calculate the squares of the side lengths. This involves simple mathematical expansion using algebra, particularly when side lengths are expressed in terms of variables. Let's consider the side \((5x + 12y)\). To find its squared length, we calculate:
\((5x + 12y)^2 = 25x^2 + 120xy + 144y^2\)
This process is repeated for each side to understand their relationships:

• \((12x + 5y)^2 = 144x^2 + 120xy + 25y^2\)
• \((13x + 13y)^2 = 169x^2 + 338xy + 169y^2\)

These calculations are essential in determining whether the triangle is right-angled, by checking if the sum of squared lengths of the two smaller sides equals the square of the longest side.

Triangle Types

Triangles are categorized based on their angles and side lengths. Understanding these types is essential in solving geometric problems.

• Right-Angled Triangle: Has a 90-degree angle and follows the Pythagorean Theorem, \(a^2 + b^2 = c^2\).
• Acute-Angled Triangle: All angles are less than 90 degrees, leading to \(a^2 + b^2 > c^2\).
• Obtuse-Angled Triangle: One angle is greater than 90 degrees, which results in \(a^2 + b^2 < c^2\).
• Equilateral Triangle: All sides and angles are equal, with each angle measuring 60 degrees.

In the given exercise, by checking the condition \(a^2 + b^2 = c^2\), we found the triangle to be right-angled, confirming its type among the listed categories.