A Tale of Three Intersecting Lines | IT

Question 17

Further, for a given set of lengths, is it possible to identify which lengths will immediately be less than the sum of the other two, without calculations?

[Hint: Consider the direct lengths in the increasing order.]

Solution

Ordering lengths helps identify which are less than the sum of others.

Step 1 — Understand the triangle rule

We are checking a rule for triangles. Any two sides' sum must be greater than the third side. Let us call the three lengths $a$ , $b$ , and $c$ . We need to check three conditions. Is $a$ less than $b+c$ ? Is $b$ less than $a+c$ ? Is $c$ less than $a+b$ ? We want to know which of these are always true.

Step 2 — Order the lengths

Let us arrange the lengths in increasing order. This means from smallest to largest. Let the smallest length be $x$ . Let the middle length be $y$ . Let the largest length be $z$ . So, we have $x \le y \le z$ .

Step 3 — Check the smallest length

Consider the smallest length, $x$ . We want to see if $x < y + z$ . The lengths $y$ and $z$ are positive. Their sum $y+z$ will always be larger than $x$ . For example, let $x=5$ , $y=10$ , $z=20$ . $5 < 10 + 20$ $5 < 30$

$\boxed{\text{True}}$ This is always true. The smallest length is always less than the sum of others.

Step 4 — Check the middle length

Consider the middle length, $y$ . We want to see if $y < x + z$ . The lengths $x$ and $z$ are positive. Their sum $x+z$ will always be larger than $y$ . For example, let $x=5$ , $y=10$ , $z=20$ . $10 < 5 + 20$ $10 < 25$

$\boxed{\text{True}}$ This is always true. The middle length is always less than the sum of others.

Step 5 — Check the largest length

Consider the largest length, $z$ . We want to see if $z < x + y$ . This condition is not always true. For example, let $x=5$ , $y=10$ , $z=20$ . $20 < 5 + 10$ $20 < 15$

$\boxed{\text{False}}$ The largest length (20 mm) is not less than $x+y$ . For another example, let $x=3$ , $y=4$ , $z=5$ . $5 < 3 + 4$ $5 < 7$ $\boxed{\text{True}}$ The largest length might or might not be less than $x+y$ .

Step 6 — Identify the lengths

If we order the lengths from smallest to largest: The smallest length is always less than the sum of others. The middle length is always less than the sum of others. The largest length is the only one to check. It might not be less than the sum of $x$ and $y$ . So, we can immediately identify the two smaller lengths.

Let us use the first example given. Lengths are 5 mm, 10 mm, 20 mm. We order them: 5 mm, 10 mm, 20 mm. The smallest length is 5 mm. The middle length is 10 mm. These two are always less than the sum of others. The largest length is 20 mm. This one is not less than $5 + 10 = 15$ .

Let us use the second example given. Lengths are 12 cm, 20 cm, 40 cm. We order them: 12 cm, 20 cm, 40 cm. The smallest length is 12 cm. The middle length is 20 cm. These two are always less than the sum of others. The largest length is 40 cm. This one is not less than $12 + 20 = 32$ .

Diagram 1

Answer

1. Yes, this always happens for the two smaller lengths. For example: (i) For 5 mm, 10 mm, 20 mm: The lengths 5 mm and 10 mm satisfy the condition. The length 20 mm does not satisfy it. (ii) For 12 cm, 20 cm, and 40 cm: The lengths 12 cm and 20 cm satisfy the condition. The length 40 cm does not satisfy it. 2. Yes, it is possible to identify which lengths will immediately be less than the sum of the other two, if we take the direct lengths in increasing order.

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Q17

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