Consider the following inequalities in respect of any triangle ABC:1. AC - AB

Question

Consider the following inequalities in respect of any triangle ABC:1. AC - AB < BC2. BC - AC < AB3. AB - BC < ACWhich of the above are correct?

Prepp Team · Accepted Answer

Understanding Triangle Inequalities

The question asks us to identify which of the given inequalities hold true for any triangle ABC. These inequalities relate the lengths of the sides of a triangle. Let the lengths of the sides BC, AC, and AB be denoted by \(a\), \(b\), and \(c\) respectively.

The Triangle Inequality Theorem

A fundamental principle in geometry is the Triangle Inequality Theorem. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For triangle ABC with side lengths \(a\), \(b\), and \(c\), the theorem can be written as:

\(a + b > c\)
\(a + c > b\)
\(b + c > a\)

Difference Inequality derived from Triangle Inequality

A direct consequence of the Triangle Inequality Theorem is that the difference between the lengths of any two sides of a triangle must be less than the length of the third side. We can see this by rearranging the basic inequalities.

Consider the inequality \(a + c > b\). If we subtract \(c\) from both sides, we get \(a > b - c\). This tells us that side \(a\) (BC) is greater than the difference between side \(b\) (AC) and side \(c\) (AB). This can also be written as \(b - c < a\).

Similarly, from \(b + c > a\), subtracting \(c\) from both sides gives \(b > a - c\), or \(a - c < b\). And from \(a + b > c\), subtracting \(b\) from both sides gives \(a > c - b\), or \(c - b < a\).

We must also consider the absolute difference, because the order of subtraction matters for the sign. For example, \(|b-c| < a\). However, the inequalities given in the question do not use absolute values. Let's analyze the given statements based on the derived inequalities \(b - c < a\), \(a - b < c\), and \(c - a < b\).

Analyzing the Given Inequalities

1. AC - AB < BC

In terms of side lengths \(a, b, c\), this inequality is \(b - c < a\).

From the Triangle Inequality Theorem, we have \(a + c > b\). Subtracting \(c\) from both sides gives \(a > b - c\). This is exactly the inequality \(b - c < a\). So, this statement is correct.

2. BC - AC < AB

In terms of side lengths \(a, b, c\), this inequality is \(a - b < c\).

From the Triangle Inequality Theorem, we have \(b + c > a\). Subtracting \(b\) from both sides gives \(c > a - b\). This is exactly the inequality \(a - b < c\). So, this statement is correct.

3. AB - BC < AC

In terms of side lengths \(a, b, c\), this inequality is \(c - a < b\).

From the Triangle Inequality Theorem, we have \(a + b > c\). Subtracting \(a\) from both sides gives \(b > c - a\). This is exactly the inequality \(c - a < b\). So, this statement is correct.

All three given inequalities are correct deductions from the Triangle Inequality Theorem.

Summary of Triangle Inequalities
Sides Compared	Sum Inequality	Difference Inequality	Corresponding Given Inequality
AB, AC vs BC (\(c, b\) vs \(a\))	\(c + b > a\)	\(\|c - b\| < a\) or \(c - b < a\) and \(b - c < a\)	AC - AB < BC (\(b - c < a\)) - Statement 1
BC, AC vs AB (\(a, b\) vs \(c\))	\(a + b > c\)	\(\|a - b\| < c\) or \(a - b < c\) and \(b - a < c\)	BC - AC < AB (\(a - b < c\)) - Statement 2
AB, BC vs AC (\(c, a\) vs \(b\))	\(c + a > b\)	\(\|c - a\| < b\) or \(c - a < b\) and \(a - c < b\)	AB - BC < AC (\(c - a < b\)) - Statement 3

Based on the analysis, all three inequalities (1, 2, and 3) are correct for any triangle ABC.

Revision Table: Triangle Side Relationships

Theorem	Relationship	Explanation
Triangle Inequality (Sum Form)	Sum of any two sides > Third side	Example: \(AB + AC > BC\) (\(c + b > a\))
Triangle Inequality (Difference Form)	Difference of any two sides < Third side	Example: \(\|AC - AB\| < BC\) (\(\|b - c\| < a\)) which implies \(b-c < a\) and \(c-b < a\).

Additional Information on Triangle Properties

The triangle inequality is a fundamental concept that helps define what can form a triangle. If the sum of the lengths of two sides is equal to or less than the length of the third side, the three segments cannot form a triangle; they would either form a degenerate triangle (collinear points) or not connect at all.

The difference form of the triangle inequality, \(|x - y| < z\), where \(x, y, z\) are side lengths, is often taught as a combined rule. This single inequality \(|x - y| < z\) is equivalent to stating both \(x - y < z\) and \(y - x < z\). The statements in the question are specific forms of the difference inequality focusing on one order of subtraction, which are directly derived from the sum form.