Context: Does a triangle exist with sidelengths , and ? This satisfies the triangle inequality: Q. Why do we not need to check the other two sides?

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Question 19

Context: Does a triangle exist with sidelengths $4\text{ cm}$ , $5\text{ cm}$ and $8\text{ cm}$ ? This satisfies the triangle inequality: $8 < 4 + 5 = 9$

Q. Why do we not need to check the other two sides?

Solution

We need to understand the triangle inequality rule.

Step 1 — Recall the Triangle Inequality

Let us call the three side lengths $a$ , $b$ , and $c$ . The triangle inequality has three rules. Each rule checks if two sides are longer than the third. Let us write these rules down.

$a + b > c$

$a + c > b$

$b + c > a$

We have side lengths of 4 cm, 5 cm, and 8 cm. Let us assign these to $a$ , $b$ , and $c$ . Let $a = \mathbf{4 \text{ cm}}$ . Let $b = \mathbf{5 \text{ cm}}$ . Let $c = \mathbf{8 \text{ cm}}$ .

Now we check each rule.

First, we check if $a + b$ is greater than $c$ . This means we check 4 cm plus 5 cm. Is this sum greater than 8 cm?

$4 + 5$

$= 9$

$\boxed{9 \text{ cm}}$

So, 9 cm is greater than 8 cm. The first rule is satisfied.

Next, we check if $a + c$ is greater than $b$ . This means we check 4 cm plus 8 cm. Is this sum greater than 5 cm?

$4 + 8$

$= 12$

$\boxed{12 \text{ cm}}$

So, 12 cm is greater than 5 cm. The second rule is satisfied.

Finally, we check if $b + c$ is greater than $a$ . This means we check 5 cm plus 8 cm. Is this sum greater than 4 cm?

$5 + 8$

$= 13$

$\boxed{13 \text{ cm}}$

So, 13 cm is greater than 4 cm. The third rule is satisfied. All three rules are satisfied. So, a triangle with these sides exists.

Diagram 1

Step 2 — The Key Insight

Let us look at the side lengths again. They are 4 cm, 5 cm, and 8 cm. The longest side is 8 cm. The two shorter sides are 4 cm and 5 cm.

The triangle inequality has a key condition. The sum of the two shorter sides is important. This sum must be greater than the longest side.

Let us call the shortest side $S_1$ . Let us call the middle side $S_2$ . Let us call the longest side $L$ .

We need to check if $S_1 + S_2 > L$ . For our triangle, this is 4 cm + 5 cm > 8 cm. This gives 9 cm > 8 cm, which is true.

Now consider the other two conditions. One condition is $S_1 + L > S_2$ . We know that $L$ is already greater than $S_2$ . Also, $S_1$ is a positive length. So, $S_1 + L$ will always be greater than $S_2$ .

The last condition is $S_2 + L > S_1$ . We know that $L$ is already greater than $S_1$ . Also, $S_2$ is a positive length. So, $S_2 + L$ will always be greater than $S_1$ .

This means we only need to check one condition. We only check if $S_1 + S_2 > L$ . The other two conditions will always be true. This happens because $L$ is the biggest side. Adding any positive side to $L$ makes the sum larger. This sum will always be greater than $S_1$ or $S_2$ .

Answer

(i) We do not need to check the other two sides because if the sum of the two shorter sides is greater than the longest side, the other two conditions are automatically satisfied.

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