A triangle has side lengths measuring [tex]$3x$[/tex] cm, [tex]$7x$[/tex] cm, and [tex]$h$[/tex] cm. Which expression describes the possible values of [tex]$h$[/tex], in cm?
A. [tex]$4x < h < 10x$[/tex]
B. [tex]$10x < h < 4x$[/tex]
C. [tex]$h = 4x$[/tex]
D. [tex]$h = 10x$[/tex]
Answer (2)
Apply the triangle inequality theorem: The sum of any two sides of a triangle must be greater than the third side.
Set up inequalities: h"> 3 x + 7 x > h , 7x"> 3 x + h > 7 x , and 3x"> 7 x + h > 3 x .
Simplify the inequalities: Obtain h < 10 x , 4x"> h > 4 x , and -4x"> h > − 4 x .
Combine the inequalities to define the range for h : 4 x < h < 10 x .
Explanation
Problem Analysis and Strategy Let's analyze the given problem. We have a triangle with side lengths 3 x , 7 x , and h . We need to find the possible values of h . To do this, we'll use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Applying the Triangle Inequality Theorem Applying the triangle inequality theorem, we can set up three inequalities:
h"> 3 x + 7 x > h
7x"> 3 x + h > 7 x
3x"> 7 x + h > 3 x
Simplifying the Inequalities Now, let's simplify each inequality:
h"> 10 x > h or h < 10 x
7x - 3x"> h > 7 x − 3 x or 4x"> h > 4 x
3x - 7x"> h > 3 x − 7 x or -4x"> h > − 4 x
Since h represents a side length, it must be positive. Also, x is a positive value since it is a scaling factor for the side lengths. Therefore, -4x"> h > − 4 x is always true if 0"> h > 0 .
Combining the Inequalities Combining the relevant inequalities, we have 4 x < h < 10 x . This means that h must be greater than 4 x and less than 10 x .
Final Answer Therefore, the expression that describes the possible values of h is 4 x < h < 10 x .
Examples
The triangle inequality is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure, the lengths of the sides must satisfy the triangle inequality to ensure the structure is stable. If the inequality is not satisfied, the structure will not be able to form a triangle and will collapse. Similarly, in navigation, the shortest distance between two points is a straight line, which corresponds to one side of a triangle. The sum of the lengths of any other two sides will always be greater than the straight-line distance.
The possible values of the side length h in the triangle must satisfy the inequality 4 x < h < 10 x . This is derived using the triangle inequality theorem. Thus, the correct answer is option A: 4 x < h < 10 x .
;
