Answered

From everyday questions to specialized queries, IDNLearn.com has the answers. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.

Type the correct answer in each box. Use numerals instead of words.

An A-frame restaurant is shaped as a triangle with two side lengths of 20 m and 30 m. Complete the inequality below to describe the range of possible lengths [tex]$x$[/tex] of the third side of the restaurant.

[tex]\[ \square \ \textless \ x \ \textless \ \square \][/tex]

---

To determine the values to fill in the boxes, you can use the triangle inequality theorem, which states:

[tex]\[ a + b \ \textgreater \ c \][/tex]
[tex]\[ a + c \ \textgreater \ b \][/tex]
[tex]\[ b + c \ \textgreater \ a \][/tex]

For side lengths 20 m and 30 m, we have:

[tex]\[ 20 + 30 \ \textgreater \ x \implies x \ \textless \ 50 \][/tex]
[tex]\[ 20 + x \ \textgreater \ 30 \implies x \ \textgreater \ 10 \][/tex]
[tex]\[ 30 + x \ \textgreater \ 20 \implies x \ \textgreater \ -10 \][/tex] (This inequality is always true since [tex]x[/tex] must be positive)

So the correct inequality describing the range of possible lengths [tex]x[/tex] of the third side is:

[tex]\[ 10 \ \textless \ x \ \textless \ 50 \][/tex]

Therefore, the boxes should be filled as follows:

[tex]\[ 10 \ \textless \ x \ \textless \ 50 \][/tex]

Sagot :

GinnyAnsweridn
To determine the range of possible lengths [tex]\( x \)[/tex] of the third side of a triangle with side lengths of 20 meters and 30 meters, we use the triangle inequality theorem. The 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.

This gives us three inequalities:
1. The sum of the first two sides must be greater than the third side:
[tex]\[ 20 + 30 > x \][/tex]
2. The sum of the first side and the unknown side must be greater than the second side:
[tex]\[ 20 + x > 30 \][/tex]
3. The sum of the second side and the unknown side must be greater than the first side:
[tex]\[ 30 + x > 20 \][/tex]

Now, let's simplify each of these inequalities:

1. Simplify the first inequality:
[tex]\[ 50 > x \][/tex]
This can be rewritten as:
[tex]\[ x < 50 \][/tex]

2. Simplify the second inequality:
[tex]\[ 20 + x > 30 \][/tex]
[tex]\[ x > 10 \][/tex]

3. Simplify the third inequality:
[tex]\[ 30 + x > 20 \][/tex]
[tex]\[ x > -10 \][/tex]
Since side lengths cannot be negative, this inequality doesn't restrict our solution further.

Combining the valid inequalities from above, we get:
[tex]\[ 10 < x < 50 \][/tex]

Thus, the range of possible lengths [tex]\( x \)[/tex] of the third side of the triangle is:
[tex]\[ 10 < x < 50 \][/tex]

Therefore, the completed inequality is:

[tex]\[ 10 < x < 50 \][/tex]
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.