Determine if it is possible to form a triangle with side lengths 7 yd, 8 yd, and 9 yd.
Understand the Problem
The question asks whether a triangle can be formed with sides of length 7 yards, 8 yards, and 9 yards. To determine this, we need to check if the sum of any two sides is greater than the third side. This is known as the triangle inequality theorem.
Answer
Yes.
Answer for screen readers
Yes, a triangle can be formed with sides of length 7 yards, 8 yards, and 9 yards.
Steps to Solve
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Check if the sum of the two smaller sides is greater than the longest side. We have sides of length 7, 8, and 9. We need to check if $7 + 8 > 9$.
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Calculate the sum of the two smaller sides. $7 + 8 = 15$
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Compare the sum to the longest side. Since $15 > 9$, the condition is satisfied for this combination of sides.
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Check all three combinations to rigorously prove the triangle inequality theorem. We need to verify that the sum of every two sides is greater than the remaining side: $7 + 8 > 9$ $7 + 9 > 8$ $8 + 9 > 7$
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Evaluate each inequality. $7 + 8 = 15 > 9$ (True) $7 + 9 = 16 > 8$ (True) $8 + 9 = 17 > 7$ (True)
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Determine if a triangle can be formed. Since all three inequalities hold true, a triangle can be formed with sides of length 7 yards, 8 yards, and 9 yards.
Yes, a triangle can be formed with sides of length 7 yards, 8 yards, and 9 yards.
More Information
The triangle inequality theorem is a fundamental concept in geometry that dictates the possible side lengths of a triangle. It ensures that the shortest distance between two points is a straight line.
Tips
A common mistake is to only check if the sum of the two smaller sides is greater than the largest side. While this is often sufficient, it's important to understand that all three combinations of sides must satisfy the triangle inequality theorem for the triangle to be valid.
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