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Answer :
To determine the range of values for the third side of a triangle when two sides are given, we use the Triangle Inequality Theorem. This theorem states that the length of any side of a triangle must be greater than the difference of the other two sides and less than the sum of the other two sides.
Given:
- First side of the triangle = 143
- Second side of the triangle = 402
Let's find the range for the third side, which we'll call [tex]\( x \)[/tex].
1. Calculate the lower bound:
For the triangle inequality to hold, the third side must be greater than the absolute difference between the two given sides:
[tex]\[
x > |143 - 402|
\][/tex]
Compute the difference:
[tex]\[
|143 - 402| = 259
\][/tex]
So, the third side must be greater than 259.
2. Calculate the upper bound:
Similarly, the third side must also be less than the sum of the two given sides:
[tex]\[
x < 143 + 402
\][/tex]
Compute the sum:
[tex]\[
143 + 402 = 545
\][/tex]
So, the third side must be less than 545.
Combining both conditions, the range of possible values for the third side [tex]\( x \)[/tex] is:
[tex]\[
259 < x < 545
\][/tex]
Therefore, the number that belongs in the green box is 259.
Given:
- First side of the triangle = 143
- Second side of the triangle = 402
Let's find the range for the third side, which we'll call [tex]\( x \)[/tex].
1. Calculate the lower bound:
For the triangle inequality to hold, the third side must be greater than the absolute difference between the two given sides:
[tex]\[
x > |143 - 402|
\][/tex]
Compute the difference:
[tex]\[
|143 - 402| = 259
\][/tex]
So, the third side must be greater than 259.
2. Calculate the upper bound:
Similarly, the third side must also be less than the sum of the two given sides:
[tex]\[
x < 143 + 402
\][/tex]
Compute the sum:
[tex]\[
143 + 402 = 545
\][/tex]
So, the third side must be less than 545.
Combining both conditions, the range of possible values for the third side [tex]\( x \)[/tex] is:
[tex]\[
259 < x < 545
\][/tex]
Therefore, the number that belongs in the green box is 259.
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