Question
Two sides of a triangle have lengths 30 cm and 82 cm . The measure of the third side is a whole number of centimeters. .
a. What is the longest the third side can be?
b. What is the shortest the third side can be?
a. The longest the third side can be is
(Simplify your answer.)
a. What is the longest the third side can be?
b. What is the shortest the third side can be?
a. The longest the third side can be is
(Simplify your answer.)
Ask by Harper Bush. in the United States
Mar 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The longest the third side can be is 111 cm.
Solution
To find the possible lengths of the third side of a triangle given two sides, we can use the triangle inequality theorem. This 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.
Let the lengths of the two sides be
a. Longest the third side can be:
According to the triangle inequality, we have:
From the first inequality:
b. Shortest the third side can be:
From the second inequality:
Summary:
- The longest the third side can be is
cm (since must be a whole number). - The shortest the third side can be is
cm.
Thus, the longest the third side can be is
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Simplify this solution Mind Expander
To determine the longest possible length of the third side in a triangle with sides measuring 30 cm and 82 cm, you use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides must be greater than the length of the remaining side. Therefore, the longest the third side can be is 30 cm + 82 cm - 1 cm = 111 cm (since it must be a whole number).
For the shortest side, the inequality states that the third side must be greater than the difference of the two sides. Thus, the shortest the third side can be is 82 cm - 30 cm + 1 cm = 53 cm (again rounding up to ensure it’s a whole number).
So to summarize:
a. The longest the third side can be is 111 cm.
b. The shortest the third side can be is 53 cm.
a. The longest the third side can be is 111 cm.
b. The shortest the third side can be is 53 cm.
