Question
Sam is drawing triangles on banners using a ruler and protractor.
Fill in the Blank Question
On one banner Sam wants to create triangles with side lengths of 9 inches and 2 inches.
How many unique (one and only one) triangles with whole number side lengths can he make?
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Answers
GPT-4o mini
To determine how many unique triangles can be created with side lengths of 9 inches and 2 inches, we need to find the possible length of the third side, which we'll denote as \( x \).
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. For sides of lengths 9 inches, 2 inches, and \( x \), we have the following inequalities:
1. \( 9 + 2 > x \)
This simplifies to \( x < 11 \).
2. \( 9 + x > 2 \)
This simplifies to \( x > -7 \). (This inequality is always satisfied since \( x \) must be a positive length.)
3. \( 2 + x > 9 \)
This simplifies to \( x > 7 \).
Combining these effective inequalities, we get:
- \( x < 11 \)
- \( x > 7 \)
Thus, \( x \) must be in the range:
\[
7 < x < 11
\]
Since \( x \) must be a whole number, the possible whole number values for \( x \) are:
- \( 8 \)
- \( 9 \)
- \( 10 \)
In total, there are 3 valid values for \( x \) that meet the conditions for forming a triangle. Therefore, Sam can create **3 unique triangles** with the given side lengths.
**Answer:** 3
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. For sides of lengths 9 inches, 2 inches, and \( x \), we have the following inequalities:
1. \( 9 + 2 > x \)
This simplifies to \( x < 11 \).
2. \( 9 + x > 2 \)
This simplifies to \( x > -7 \). (This inequality is always satisfied since \( x \) must be a positive length.)
3. \( 2 + x > 9 \)
This simplifies to \( x > 7 \).
Combining these effective inequalities, we get:
- \( x < 11 \)
- \( x > 7 \)
Thus, \( x \) must be in the range:
\[
7 < x < 11
\]
Since \( x \) must be a whole number, the possible whole number values for \( x \) are:
- \( 8 \)
- \( 9 \)
- \( 10 \)
In total, there are 3 valid values for \( x \) that meet the conditions for forming a triangle. Therefore, Sam can create **3 unique triangles** with the given side lengths.
**Answer:** 3