A triangle has side lengths measuring [tex]$3x \text{ cm}$[/tex], [tex]$7x \text{ cm}$[/tex], and [tex]$h \text{ cm}$[/tex]. Which expression describes the possible values of [tex]$h$[/tex] in cm?
A. [tex]$4x \ \textless \ h \ \textless \ 10x$[/tex]
B. [tex]$10x \ \textless \ h \ \textless \ 4x$[/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 the inequalities: h"> 3 x + 7 x > h , 7x"> 3 x + h > 7 x , and 3x"> 7 x + h > 3 x .
Simplify the inequalities: h"> 10 x > h , 4x"> h > 4 x , and -4x"> h > − 4 x .
Combine the relevant inequalities to find the range of possible values for h : 4 x < h < 10 x .
Explanation
Problem Analysis and Setup Let's analyze the problem. We are given a triangle with side lengths 3 x , 7 x , and h . We need to find the possible values of h . The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will use this theorem to set up inequalities and solve for h .
Applying Triangle Inequality Applying the triangle inequality theorem, we have the following three inequalities:
h"> 3 x + 7 x > h
7x"> 3 x + h > 7 x
3x"> 7 x + h > 3 x
Simplifying 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 length and therefore positive, so -4x"> h > − 4 x is always true. Thus, we only need to consider the first two inequalities.
Combining the Results Combining the inequalities h < 10 x and 4x"> h > 4 x , we get 4 x < h < 10 x .
Final Answer Therefore, the possible values of h are described by the expression 4 x < h < 10 x .
Examples
The triangle inequality is a fundamental concept in geometry and has many practical applications. For example, if you are building a triangular structure, such as a roof or a bridge, you need to ensure that the lengths of the sides satisfy the triangle inequality. If the inequality is not satisfied, the structure will not be stable and may collapse. Another example is in navigation. If you know the distances between three locations, you can use the triangle inequality to determine if it is possible to travel between them in a straight line.
Using the triangle inequality theorem, we find that for the triangle with sides 3 x , 7 x , and h , the possible values of h satisfy the expression 4 x < h < 10 x . Hence, the correct answer is option A.
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