Question

Multiple Choice Points \(J, K\), and \(L\) are collinear, with \(K\) between \(J\) and L. If \(K L=6 \frac{1}{3}\) and \(J L=16 \frac{2}{5}\), what is the measure of \(\overline{J K}\) ? (Lesson 2-2) (A) 10 (B) \(10 \frac{1}{15}\) (C) \(22 \frac{1}{2}\) (D) \(22 \frac{11}{15}\)

Short answer

The measure of \(\overline{J K}\) is \(10 \frac{1}{15}\).

Step-by-step solution

Understand the collinearity

Since points \(J, K,\) and \(L\) are collinear and \(K\) is positioned between \(J\) and \(L\), we can use the segment addition postulate which states that \(JK + KL = JL\).

Convert mixed numbers to improper fractions

The segment \(KL\) is given as \(6 \frac{1}{3}\), which can be converted to an improper fraction: \[ \frac{6 \times 3 + 1}{3} = \frac{19}{3} \].Similarly, \(JL\) is given as \(16 \frac{2}{5}\), which can be converted to: \[ \frac{16 \times 5 + 2}{5} = \frac{82}{5} \].

Set up the equation

Using the segment addition postulate, we can write the equation for the segments: \[ JK + KL = JL \]Substitute the known values: \[ JK + \frac{19}{3} = \frac{82}{5} \]

Solve for JK

To solve for \(JK\), we rearrange the equation: \[ JK = \frac{82}{5} - \frac{19}{3} \]Find a common denominator (which is 15) to subtract these fractions:\[ JK = \frac{246}{15} - \frac{95}{15} = \frac{151}{15} \]

Convert the result to a mixed number

Convert the improper fraction \(\frac{151}{15}\) back to a mixed number. Dividing 151 by 15 gives 10 with a remainder of 1, so:\[ JK = 10 \frac{1}{15} \].

Verify the answer

Ensure the mixed number matches one of the choices. The solution \(10 \frac{1}{15}\) matches option (B).

Key concepts

Collinear Points

Collinear points are three or more points that lie on the same straight line. In the given exercise, points \( J, K, \) and \( L \) are identified as collinear, implying they all align perfectly across a single straight path. This property is essential in geometry, as it helps us understand relations between points and segments.
When a point like \( K \) is said to be 'between' points \( J \) and \( L \), it defines a particular sequence on their path. It indicates the order and positioning, which is useful when applying the Segment Addition Postulate.
So, having collinear points guides us in forming equations that highlight these relationships. Visualizing them as dots on a line segment can make these concepts clear and straightforward.

Segment Addition Postulate

The Segment Addition Postulate is a fundamental principle in geometry. It states that if a point \( K \) lies on the line segment \( JL \) between the endpoints \( J \) and \( L \), then the sum of the lengths of the segments \( JK \) and \( KL \) is equal to the length of the entire segment \( JL \).
For this exercise, the postulate is expressed as:

• \( JK + KL = JL \)

This equation plays a critical role in solving the problem, as it helps establish the relationship needed to find the unknown length \( JK \).
By substituting known values, it transforms into a practical tool that allows easy computation of any unknown segment when the other two are known.

Fraction Conversion

Fraction conversion is a useful mathematical skill, especially when dealing with mixed numbers and improper fractions. In this exercise, such conversions were necessary to solve for \( JK \) using the Segment Addition Postulate.
To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fractional part, then add the numerator. Finally, place this sum over the original denominator.
For example, converting \(6 \frac{1}{3}\) involves:

• 6 gives \(6 \times 3 + 1 = 19\), leading to the improper fraction \(\frac{19}{3}\).

Similarly, for \(16 \frac{2}{5}\), the improper fraction is \(\frac{82}{5}\).
Converting back, for instance, \(\frac{151}{15}\) involves dividing to get \(10\) and a remainder. This results in the mixed number \(10 \frac{1}{15}\).
This understanding streamlines calculations, ensuring clarity and precision in geometric problem-solving.