Q43. - Algebra 1

Question

GEOMETRY Use the following statement. 

If there are three line-segments $\overline{AB},\overline{BC}$ and  $\overline{CD}$ then they form a triangle.

a. Draw a diagram to provide an example for the conditional statement.

b. Draw a diagram to provide a counterexample for the conditional statement.

Step-by-Step Solution

Answer

a. An example for the given conditional statement is when the points $A$ and $D$ are the same and $A, B$ and $C$ are non-collinear points. The diagram has been provided.

a. A counterexample for the given conditional statement is either when the points $A$ and $D$ are not the same or when $A, B$ and $C$ are collinear points. The diagrams have been provided.

Part a. Step 1. State the concept.

In an if-then statement, the part following “if” is the hypothesis, and the part following “then” is the conclusion.

Part a. Step 2. List the given data.

The given statement is “If there are three line-segments $\overline{AB},\overline{BC}$ and $\overline{CD}$, then they form a triangle”.

Part a. Step 3. Give example.

An example of a conditional statement is a case where the hypothesis and the conclusion are both true. Here, an example has to be a case where three line-segments $\overline{AB},\overline{BC}$ and $\overline{CD}$ form a triangle.

A triangle has three vertices. If three line segments form the three sides of a triangle then the line segments must be formed using three points. So, $A$ and $D$ must be the same point.

Moreover, three points can form a triangle only when they are non-collinear. Thus, $A, B$ and $C$ must be non-collinear.

Part a. Step 4. Diagram.

Part b. Step 1. State the concept.

In an if-then statement, the part following “if” is the hypothesis and the part following “then” is the conclusion.

Part b. Step 2. List the given data.

The given statement is “If there are three line-segments $\overline{AB},\overline{BC} \mathrm{and} \overline{CD}$, then they form a triangle”.

Part b. Step 3. Give counterexample.

A counterexample of a conditional statement is a case where the hypothesis is true but the conclusion is false. Here, a counterexample has to be a case where three line-segments $\overline{AB},\overline{BC} \mathrm{and} \overline{CD}$ does not form a triangle.

A triangle has three vertices. If three line segments are formed using four distinct points then the line segments do not form the three sides of a triangle. So, $A$ and $D$ must be distinct points.

Moreover, three points can form a triangle only when they are non-collinear. Thus, if $A, B, C$ and $D$ are collinear, then a triangle will not be formed.

Part b. Step 4. Diagram.

The two cases which serve as counterexamples to the given statement are shown in the diagrams below: -