Question

Critical Thinking In trapezoid \(A B C D\), \(A B=10, B C=23\), and \(C D=11\). What is the range of possible measures for \(\overline{A D}\) ? (Hint: First find the range of possible measures for \(\overline{A C}\).)

Short answer

The range for \(\overline{AD}\) is between 2 and 44, exclusive.

Step-by-step solution

Identify Known Dimensions and Trapezoid Characteristics

We are given a trapezoid \(ABCD\) where \(AB = 10\), \(BC = 23\), and \(CD = 11\). Trapezoids have one pair of parallel sides, but we are not provided information about which sides are parallel. The goal is to find the range of possible lengths for \(\overline{AD}\).

Apply the Triangle Inequality to Side Considerations

To determine the range of possible lengths for \(\overline{AC}\), apply the triangle inequality to \(\triangle ABC\) by considering sides \(AB\), \(BC\), and \(AC\):1. \(AC < AB + BC\) implies \(AC < 10 + 23 = 33\).2. \(AB < AC + BC\) implies \(10 < AC + 23\) resulting in \(AC > -13\), which is non-restrictive.3. \(BC < AB + AC\) implies \(23 < 10 + AC\) resulting in \(AC > 13\).Combine these inequalities to narrow down \(\overline{AC}\) to the range \(13 < AC < 33\).

Translate Range of \(AC\) to \(AD\) Using Triangle Inequality

Now, consider \(\triangle ACD\) and apply the triangle inequality with sides \(AD\), \(CD\), and \(AC\):1. \(AD < AC + CD\) implies \(AD < AC + 11\).2. \(AC < AD + CD\) implies \(AC < AD + 11\) and since \(13 < AC < 33\), consider the relevant part: \(13 < AD + 11\) resulting in \(AD > 2\).3. \(CD < AC + AD\) which is non-restrictive when combined with \(AD > 2\) and \(AC < 33\).Using \(AC < 33\), the strongest conclusion is \(AD < 33 + 11 = 44\). Combine with \(AD > 2\) to narrow down possible values.

Conclusion about \(AD\)

Combine the results from previous steps: we have established that \(AD > 2\) and \(AD < 44\). Therefore, the range for \(\overline{AD}\) is between 2 and 44, exclusive of these endpoints.

Key concepts

Trapezoids

Trapezoids are a special kind of quadrilateral, which means they have four sides. What makes trapezoids unique is that they have at least one pair of parallel sides. In a trapezoid, these parallel sides are often called the 'bases'. The other two sides, which are not parallel, are referred to as the 'legs'. In the context of solving geometric problems, knowing which sides are parallel can greatly influence the approach to finding solutions.
When dealing with trapezoids, it’s important to remember that the length of the parallel sides can help determine the height and area of the trapezoid. However, in some problems, like the one we're exploring, you may not initially know which sides are parallel. Therefore, understanding the properties of trapezoids allows us to apply broader geometric principles, such as the Triangle Inequality, to find unknown lengths or measures.
For trapezoid ABCD, knowing lengths such as AB, BC, and CD helps us use related triangles within the trapezoid, such as triangle ABC or triangle ACD, to solve problems. This understanding paves the way to effectively applying mathematical inequalities and reasoning through geometry challenges.

Triangle Inequality

The Triangle Inequality is a fundamental principle in geometry that applies to triangles. It states that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
This inequality can be broken down into three basic conditions for a triangle with sides a, b, and c:

• a + b > c
• a + c > b
• b + c > a

These inequalities help us understand the possible lengths of the sides of a triangle. In the problem involving trapezoid ABCD, we use the Triangle Inequality to assess the potential lengths of the diagonal AC.
By analyzing the relationships between the known sides of a triangle within the trapezoid (for example, triangle ABC), we can establish a range for unknown sides such as AC and, subsequently, AD. The Triangle Inequality isn't just about numbers; it's a tool for exploring and reasoning about the geometrical shapes to ensure that they make sense logically and physically.

Critical Thinking

Critical Thinking in mathematics involves more than just performing calculations. It's about understanding the concepts that underpin the calculations and applying them thoughtfully to solve problems. In the exercise with trapezoid ABCD, critical thinking is necessary to navigate through the given data and derive a solution.
To solve this problem, we first identify what we know and what we need to find. We have lengths for sides AB, BC, and CD, and we need to determine the possible range for AD. This requires a blend of geometric reasoning and application of mathematical principles like the Triangle Inequality.
Critical thinking involves asking questions like:

• How can I use the known lengths to find other unknown measures?
• What geometric principles are applicable here?
• Have I considered all possibilities and conditions?

By using critical thinking, we can break down complex problems into smaller, manageable parts and use logical reasoning to arrive at a solution. This approach transforms a potentially overwhelming task into a series of simpler problems, each solved with careful thought and analysis.