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!50 POINTS! (4 SIMPLE GEOMETRY QUESTIONS)

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User Lauran
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1 Answer

4 votes

Answer:

1 ) 7 < x < 11

2) 1 < x < 5

3) 1 < x < 11

4) 5 < x < 13

Explanation:

Triangle inequality theorem : the sum of any two sides is always greater than the third side

Let the unknown side be x nad let the known sides by a and b

We will have three inequalities:

(i) a + b > x

(ii) a + x > b

⇒ x > b - a

This will be negative if b < a

(iii) b + x > a

⇒ x > a - b

This will be negative if a < b

From (ii) and (iii)

if a < b, (b-a) will be positive and x > (b - a)

otherwise x > (a - b)

By combining, we can say that x > |a - b|

So our inequalities will be:

(a + b)> x and x > |a - b|

|a - b| < x < (a + b)

This is the required range

1) a = 9, b = 2

|a - b| < x < (a + b)

⇒ |9 - 2| < x < (9 + 2)

⇒ |7| < x < 11

7 < x < 11

2) a = 2, b = 3

|a - b| < x < (a + b)

⇒ |2 - 3| < x < (2 + 3)

⇒ |-1| < x < 5

1 < x < 5

3) a = 5, b = 6

|a - b| < x < (a + b)

⇒ |5 - 6| < x < (5 + 6)

⇒ |-1| < x < 11

1 < x < 11

4) a = 4, b = 9

|a - b| < x < (a + b)

⇒ |4 - 9| < x < (4 + 9)

⇒ |-5| < x < 13

5 < x < 13

User Juliocesar
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8.3k points

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