Question

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. $4,5,\text{9}$ c. $4,5,\text{7}$

Short answer

Option C forms a triangle using the triangle inequality theorem.

Step-by-step solution

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.

Step 2. Calculation.

As per the theorem,

$\begin{array}{c}x+y>z\\ x+z>y\\ y+z>x\end{array}$

Now check for all the three options given.

Option a- $4,5,10$

Let$x=4,y=5,z=10$.

$x+y>z$gives$4+5>10$.

Now add and check-

$9>10$, Which is incorrect.

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

Option b- $4,5,9$

Let$x=4,y=5,z=9$.

$x+y>z$gives$4+5>9$.

Now add and check-

$9>10$, Which is incorrect.

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

Option c- $4,5,\text{7}$

Let$x=4,y=5,z=7$.

$x+y>z$gives $4+5>7$.

Now add and check-

$9>7$, Which is correct.

Now with the other two equations as well.

$x+z>y$gives $4+7>5$.

Now add and check-

$11>5$, Which is correct.

$y+z>x$gives$5+7>7$.

Now add and check-

$12>5$, which is correct.

So, option c fulfills the inequality theorem.

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.