The Correct Answer and Explanation is:
The correct answer is 17 and 2/3.
This problem is solved using a fundamental concept in geometry known as the Triangle Proportionality Theorem, which is also called the intercept theorem or Thales’s theorem. This theorem describes the relationship between segments created when two parallel lines intersect a pair of transversals, which are lines that cross the parallel lines. The theorem states that the parallel lines divide the transversals into segments that are proportional to each other.
In the diagram presented, we see two transversals that intersect. These are crossed by what we can infer are two parallel line segments. The lengths of the resulting segments on the transversals are given. On one transversal, the segments have lengths of 10.6 and the unknown distance x. On the other transversal, the corresponding segments have lengths of 10.2 and 17.
Based on the Triangle Proportionality Theorem, we can set up an equation that equates the ratios of the corresponding segments:
10.6 / x = 10.2 / 17
To solve for the unknown value x, we can begin by cross-multiplying. This method helps to eliminate the fractions from the equation:
10.6 * 17 = 10.2 * x
Next, we perform the multiplication on the left side of the equation:
180.2 = 10.2 * x
To isolate x, we need to divide both sides of the equation by 10.2:
x = 180.2 / 10.2
To make this division easier, we can multiply both the numerator and the denominator by 10 to work with whole numbers:
x = 1802 / 102
This fraction can be simplified by dividing both the top and bottom by 2:
x = 901 / 51
Finally, we perform the division or convert the improper fraction to a mixed number. When 901 is divided by 51, the result is 17 with a remainder of 34. This gives us the mixed number 17 and 34/51. The fractional part, 34/51, simplifies to 2/3 because both 34 and 51 are divisible by 17.
Therefore, the exact distance for x is 17 and 2/3.
