Problem 65 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the converse and the contrapositive to the following statements. (a) (Let \(a, b\), and \(c\) be the lengths of sides of a triangle.) If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle. (b) If angle \(A B C\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\).

Step-by-Step Solution

Answer

(a) Converse: If the triangle is right, then \(a^{2}+b^{2}=c^{2}\). Contrapositive: If not right, then \(a^{2}+b^{2} \neq c^{2}\). (b) Converse: If \(0^{\circ} < \angle ABC < 90^{\circ}\), then acute. Contrapositive: If not \(0^{\circ} < \angle ABC < 90^{\circ}\), then not acute.

Step 1: Understand the Original Statement

The original statement is given in the form: If \(P\), then \(Q\). Here, (a) is: If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle. (b) is: If angle \(ABC\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\).

Step 2: Write the Converse

The converse of a statement switches the hypothesis and conclusion. (a) Converse: If the triangle is a right triangle, then \(a^{2}+b^{2}=c^{2}\). (b) Converse: If the measure of angle \(ABC\) is greater than \(0^{\circ}\) and less than \(90^{\circ}\), then angle \(ABC\) is acute.

Step 3: Write the Contrapositive

The contrapositive of a statement negates and switches the hypothesis and conclusion. (a) Contrapositive: If the triangle is not a right triangle, then \(a^{2}+b^{2} eq c^{2}\). (b) Contrapositive: If the measure of angle \(ABC\) is not greater than \(0^{\circ}\) or not less than \(90^{\circ}\), then angle \(ABC\) is not acute.

Key Concepts

Logical Reasoning

Understanding logical reasoning is fundamental in mathematics and everyday problem-solving. Logical reasoning involves drawing conclusions based on given premises or statements. It is divided into two basic forms: deductive reasoning and inductive reasoning. Deductive reasoning applies general rules to specific cases to arrive at a conclusion, while inductive reasoning makes broad generalizations from specific observations. In the context of statements such as 'If \(P\), then \(Q\)', **logical reasoning** helps us to understand related statements like the converse and contrapositive.

**Converse:** This statement flips the hypothesis and conclusion of the original statement. For example, from the original statement 'If \(a^{2} + b^{2} = c^{2}\), then the triangle is a right triangle', the converse would be 'If the triangle is a right triangle, then \(a^{2} + b^{2} = c^{2}\)'.

**Contrapositive:** This negates and switches the hypothesis and conclusion. So for the original geometric statement, the contrapositive is 'If the triangle is not a right triangle, then \(a^{2} + b^{2} eq c^{2}\)'.

These logical manipulations help in analyzing mathematical statements, demonstrating that they hold in various scenarios.

Geometry

Geometry is a branch of mathematics that explores the properties and relationships of points, lines, surfaces, and shapes. In many geometric problems, understanding the properties of figures like triangles is crucial. The Pythagorean theorem, for instance, is a fundamental concept in geometry, stating that in a right triangle, the square of the hypotenuse length (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)).

In the statement 'If \(a^{2} + b^{2} = c^{2}\), then the triangle is a right triangle', we see a direct application of this theorem. The converse of this theorem is also true, which is a significant property, meaning if a triangle is right, then \(a^{2} + b^{2}\) will indeed equal \(c^{2}\). The **converse** demonstrates the geometric certainty within these specific conditions.

Another important aspect is understanding angles in triangles, such as acute angles, which are less than \(90^{\circ}\). So, in the statement, 'If angle \(ABC\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\)', students comprehend how the constraints define the type of angle being measured.

Analytical Thinking

Analytical thinking is about breaking down complex problems into smaller, more manageable parts, which can be examined and understood. It is crucial for problem-solving in mathematics and real-world scenarios.

When approaching exercises involving statements and their truth values, **analytical thinking** helps determine the validity of the converse and contrapositive. This skill aids students in understanding not just surface-level ideas but the underlying logical structure. For instance, considering whether the converse or contrapositive of a statement maintains the same truth value as the original statement requires keen observation and logical dissection.

By dissecting statements like "If angle \(ABC\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\)", analytical thinkers can analyze the necessary and sufficient conditions for a condition to hold. If one part of a statement holds true, analytical thinkers can verify its implications on the truth of other related statements. This form of thinking empowers students to approach mathematical problems more effectively and apply these skills beyond just theoretical exercises.