To determine the type of triangle based on its side lengths, we can use the triangle inequality theorem and the Pythagorean theorem.
Given side lengths:
a = 14
b = 48
c = 50
1. Triangle Inequality Theorem:
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Using the triangle inequality theorem, we can check if the given side lengths satisfy this condition:
a + b > c
14 + 48 > 50
62 > 50 (True)
b + c > a
48 + 50 > 14
98 > 14 (True)
a + c > b
14 + 50 > 48
64 > 48 (True)
Since all three inequalities are true, the given side lengths form a valid triangle.
2. Pythagorean Theorem:
Next, we can use the Pythagorean theorem to determine the type of triangle.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
a^2 + b^2 = c^2
14^2 + 48^2 = 50^2
196 + 2304 = 2500
2500 = 2500 (True)
Since the equation is true, the given triangle is a right triangle.
In conclusion, the triangle with side lengths 14, 48, and 50 units is a right triangle.