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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Using this theorem, determine if the given side lengths for a triangle. Explain your reasoning.

  1. 4,5,10 b. 4,5,9 c. 4,5,7

Short Answer

Expert verified

Option C forms a triangle using the triangle inequality theorem.

Step by step solution

01

Step 1. Given Information.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

02

Step 2. Calculation.

As per the theorem,

x+y>zx+z>yy+z>x

Now check for all the three options given.

Option a- 4,5,10

Letx=4,y=5,z=10.

x+y>zgives4+5>10.

Now add and check-

9>10, Which is incorrect.

So, option a cannot form a triangle using the inequality theorem.

Option b- 4,5,9

Letx=4,y=5,z=9.

x+y>zgives4+5>9.

Now add and check-

9>10, Which is incorrect.

So, option b cannot form a triangle using the inequality theorem.

Option c- 4,5,7

Letx=4,y=5,z=7.

x+y>zgives 4+5>7.

Now add and check-

9>7, Which is correct.

Now with the other two equations as well.

x+z>ygives 4+7>5.

Now add and check-

11>5, Which is correct.

y+z>xgives5+7>7.

Now add and check-

12>5, which is correct.

So, option c fulfills the inequality theorem.

03

Step 3. Conclusion.

Since option C fulfils the sum of the lengths of any two sides of a triangle which is greater than the length of the third side.

Hence, option C forms a triangle using inequality theorem.

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