College

Given 143 and 402 as the lengths of two sides of a triangle, find the range of values for the third side.

[tex] 402 - 143 \ < \ x \ < \ 402 + 143 [/tex]

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.

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