Question

A land surveyor must calculate the distance across a river, segment k l, in feet. the surveyor measures the distance of segment j k, segment k n, and segment l m. triangle j l m overlaps triangle j k n, where side k n measures 18 feet and is parallel to l m, and side j k measures 21 feet. triangle j l m has a right angle l, and side length l m has a measure of 48 feet. given that segment k n is parallel to segment l m, what is the measure, in feet, of the distance across the river? a) 56 feet b) 77 feet c) 35 feet d) 48 feet

Answer 1

The distance across the river, represented by segment KL, is 56 feet which is calculated using the properties of similar triangles and the given measurements.

Step-by-step explanation:

The land surveyor is required to calculate the distance across a river, which is represented by segment KL. Given the measurements of the adjacent segments and the fact that segment KN is parallel to segment LM, we can utilize geometric properties to find the required distance. Since triangles JKN and JLM are similar (one pair of parallel sides and all corresponding angles are equal in similar triangles), the sides are proportional. The proportion can be set up as follows:

JK / JL = KN / LM

We know JK = 21 feet, KN = 18 feet, and LM = 48 feet. We can substitute the known lengths into the proportion:

21 / JL = 18 / 48

By solving for JL, we find:

JL = 21 × (48 / 18)

JL = 21 × (8 / 3)

JL = 56 feet

Therefore, the measure of the distance across the river, segment KL, is 56 feet.

Answer 2

The answer is (a) 56 feet.

1. Identifying relevant information:

We know the following values:

JK = 21 feet

KN = 18 feet (parallel to LM)

LM = 48 feet (right angle at L)

2. Applying triangle relationships:

Triangle JKN: Since JN is parallel to LM, triangles JKN and JLM are similar (AA Similarity).

Proportionality in similar triangles: Corresponding sides in similar triangles are proportional. Therefore, KN/JM = JK/JL. Substituting known values: 18/48 = 21/JL.

3. Solving for JL:

Cross-multiplying the equation from step 2: 18 * JL = 21 * 48. Simplifying: JL = (21 * 48) / 18 = 56 feet.

4. Determining distance across the river:

JL represents the distance across the river (KL).

Summary:

By analyzing the similarities between the overlapping triangles and using the proportionality of corresponding sides, we can calculate the distance across the river as 56 feet.