Question

an acute triangle has two sides measuring 8 cm and 10 cm. what is the best representation of the possible range of values for the third side, s?

• ( 2 < s < 10 )
• ( 6 < s < 12.8 )
• ( s < 2 ) or ( s > 10 )
• ( s < 6 ) or ( s > 12.8 )

Explanation:

Step1: Recall Triangle Inequality Theorem

For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Also, for an acute triangle, we use the Pythagorean inequality. Let's first apply the basic triangle inequality. The difference between the two given sides is \(10 - 8=2\) and the sum is \(10 + 8 = 18\). So by triangle inequality, \(2<s<18\). But since it's an acute triangle, we need to check the Pythagorean inequality. If \(s\) is the longest side, then \(8^{2}+10^{2}>s^{2}\), so \(s^{2}<164\), \(s < \sqrt{164}\approx12.8\). If \(10\) is the longest side, then \(8^{2}+s^{2}>10^{2}\), so \(s^{2}>36\), \(s > 6\). But the first option is \(2 < s < 18\) (wait, no, let's re - check the options. Wait, the options are: 1. \(2 < s < 18\); 2. \(6 < s < 12.8\); 3. \(s < 2\) or \(s>18\); 4. \(s < 6\) or \(s>12.8\). Wait, maybe I misread the options. Let's re - evaluate.

First, basic triangle inequality: for a triangle with sides \(a,b,c\), \(|a - b|<c<a + b\). Here \(a = 8\), \(b = 10\), so \(|10 - 8|=2\) and \(10 + 8=18\), so \(2 < s < 18\) from basic triangle inequality. But for acute triangle:

Case 1: \(s\) is the longest side (\(s\geq10\)): Then \(8^{2}+10^{2}>s^{2}\), \(64 + 100>s^{2}\), \(s^{2}<164\), \(s<\sqrt{164}\approx12.8\).

Case 2: \(10\) is the longest side (\(s < 10\)): Then \(8^{2}+s^{2}>10^{2}\), \(64+s^{2}>100\), \(s^{2}>36\), \(s > 6\).

Combining these two cases with the basic triangle inequality, we get \(6 < s < 12.8\). But the first option is \(2 < s < 18\) (which is the basic triangle inequality range). Wait, maybe the question has a typo or I misread. Wait, the original options: the first option is \(2 < s < 18\), the second is \(6 < s < 12.8\), the third is \(s < 2\) or \(s>18\), the fourth is \(s < 6\) or \(s>12.8\).

Wait, the basic triangle inequality gives \(2 < s < 18\) (since the sum of two sides must be greater than the third, and the difference must be less than the third). The other options: \(s < 2\) or \(s>18\) would not form a triangle. \(s < 6\) or \(s>12.8\) is for non - acute (obtuse) or non - triangle. The option \(6 < s < 12.8\) is for acute, but the first option \(2 < s < 18\) is the range for a valid triangle (including acute, right, and obtuse). But the question says "an acute triangle", but maybe the options are mis - printed. Wait, maybe the first option is the triangle inequality range (valid triangle), and the others are for acute or obtuse. Wait, no, the question is asking for the possible range of the third side for an acute triangle. But let's check the options again.

Wait, the first option is \(2 < s < 18\), which is the range for a triangle (by triangle inequality). The option \(s < 2\) or \(s>18\) would not form a triangle (since the sum of two sides must be greater than the third). The option \(s < 6\) or \(s>12.8\) would be for obtuse triangles (because if \(s < 6\), then \(8^{2}+s^{2}<10^{2}\) (obtuse with 10 as the longest side), and if \(s>12.8\), then \(8^{2}+10^{2}<s^{2}\) (obtuse with \(s\) as the longest side)). The option \(6 < s < 12.8\) is for acute. But maybe the question has a mistake, or I misread the options. Wait, the first option is \(2 < s < 18\) (triangle inequality), the second is \(6 < s < 12.8\) (acute triangle). But let's check the original problem again. The problem says "an acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, s?".

Wait, maybe the options are:

1. \(2 < s < 18\)

2. \(6 < s < 12.8\)

3. \(s < 2\) or \(s>18\)

4. \(s < 6\) or \(s>12.8\)

The basic triangle inequality gives \(2 < s < 18\) (for any triangle). For an acute triangle, we need \(6 < s < 12.8\). But maybe the question's options are different. Wait, the user's options:

- \(2 < s < 18\)

- \(6 < s < 12.8\)

- \(s < 2\) or \(s>18\)

- \(s < 6\) or \(s>12.8\)

The correct range for a triangle (any triangle) is \(2 < s < 18\) (by triangle inequality: the length of a side must be greater than the difference of the other two sides and less than the sum of the other two sides). The options \(s < 2\) or \(s>18\) are for non - triangle (since if \(s\leq2\), \(8 + s\leq10\) (when \(s = 2\), \(8+2 = 10\), which is a degenerate triangle), and if \(s\geq18\), \(8 + 10\leq s\) (when \(s = 18\), \(8 + 10=18\), degenerate triangle)). The options \(s < 6\) or \(s>12.8\) are for obtuse triangles (as we saw from Pythagorean inequality). The option \(6 < s < 12.8\) is for acute triangles. But maybe the question is just asking for the triangle inequality range (not considering acute for a moment, but the problem says acute). Wait, maybe there is a mistake in the problem, or I misread. Wait, the first option is \(2 < s < 18\), which is the range for a valid triangle (including acute, right, obtuse). The other options: \(s < 2\) or \(s>18\) is invalid, \(s < 6\) or \(s>12.8\) is obtuse, \(6 < s < 12.8\) is acute. But the problem says "an acute triangle", so the correct range is \(6 < s < 12.8\). But let's check the original options again. The user's options:

First option: \(2 < s < 18\)

Second option: \(6 < s < 12.8\) (maybe written as \(6 < s < 12.8\))

Third option: \(s < 2\) or \(s>18\)

Fourth option: \(s < 6\) or \(s>12.8\)

So the correct answer for the range of the third side of an acute triangle with sides 8 and 10 is \(6 < s < 12.8\). But if we consider the triangle inequality (without considering acute), it's \(2 < s < 18\). But the problem specifies acute. Wait, maybe the question has a typo, and the first option is the triangle inequality range. Let's re - check the triangle inequality: for a triangle with sides \(a,b,c\), \(|a - b|<c<a + b\). So \(|10 - 8| = 2\) and \(10+8 = 18\), so \(2 < s < 18\). This is the range for a valid triangle (non - degenerate). So if the question is about a triangle (not necessarily acute), the answer is \(2 < s < 18\). But the problem says acute. However, maybe the options are wrong, or I made a mistake.

Wait, let's calculate the Pythagorean inequality properly.

If the triangle is acute:

- If \(s\) is the longest side (\(s\geq10\)): \(a^{2}+b^{2}>c^{2}\), where \(a = 8\), \(b = 10\), \(c = s\). So \(8^{2}+10^{2}>s^{2}\), \(64 + 100>s^{2}\), \(s^{2}<164\), \(s<\sqrt{164}\approx12.8\). Also, \(s\geq10\) (since it's the longest side), so \(10\leq s<12.8\).

- If \(10\) is the longest side (\(s < 10\)): \(a^{2}+c^{2}>b^{2}\), where \(a = 8\), \(c = s\), \(b = 10\). So \(8^{2}+s^{2}>10^{2}\), \(s^{2}>100 - 64=36\), \(s > 6\). Also, \(s < 10\) (since 10 is the longest side), so \(6 < s < 10\).

Combining these two cases, we get \(6 < s < 12.8\).

Now, let's check the options:

- Option 1: \(2 < s < 18\): This is the range for any triangle (including non - acute).

- Option 2: \(6 < s < 12.8\): This is the range for an acute triangle.

- Option 3: \(s < 2\) or \(s>18\): This is for non - triangle (degenerate or impossible).

- Option 4: \(s < 6\) or \(s>12.8\): This is for obtuse triangle.

So the correct answer for an acute triangle is \(6 < s < 12.8\). But maybe the first option is a trick, but according to the Pythagorean inequality for acute triangles, the range is \(6 < s < 12.8\). Wait, but the first option is \(2 < s < 18\) (triangle inequality). Maybe the problem is not about acute, but the user made a mistake. But based on the problem statement "an acute triangle", the correct range is \(6 < s < 12.8\). But let's check the options again. If the second option is \(6 < s < 12.8\), then that's the answer. But if the first option is \(2 < s < 18\), which is the triangle inequality, maybe the problem has a mistake.

Wait, maybe I misread the options. Let's assume that the options are:

A. \(2 < s < 18\)

B. \(6 < s < 12.8\)

C. \(s < 2\) or \(s>18\)

D. \(s < 6\) or \(s>12.8\)

The correct answer for the range of the third side of an acute triangle with sides 8 and 10 is B. \(6 < s < 12.8\). But if we consider the triangle inequality (without acute), it's A. But the problem says acute. So the answer should be B. But let's confirm with the triangle inequality and Pythagorean inequality.

Answer:

B. \(6 < s < 12.8\)