Consider a segment with endpoints $S(-7,-6)$ and $T(2,4)$.
What is the length of $\overline{ ST }$ ?
Answer (1)
Identify the coordinates of the endpoints S ( − 7 , − 6 ) and T ( 2 , 4 ) .
Apply the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates into the formula: d = ( 2 − ( − 7 ) ) 2 + ( 4 − ( − 6 ) ) 2 = 9 2 + 1 0 2 .
Calculate the length: d = 181 ≈ 13.45 .
181
Explanation
Problem Analysis We are given the coordinates of the endpoints of a segment, S ( − 7 , − 6 ) and T ( 2 , 4 ) . Our goal is to find the length of the segment ST . We can use the distance formula to find the length of the segment.
Distance Formula The distance formula is given by: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the endpoints of the segment.
Applying the Formula Substitute the coordinates of points S and T into the distance formula: d = ( 2 − ( − 7 ) ) 2 + ( 4 − ( − 6 ) ) 2 d = ( 2 + 7 ) 2 + ( 4 + 6 ) 2 d = ( 9 ) 2 + ( 10 ) 2 d = 81 + 100 d = 181 d ≈ 13.45
Final Answer The length of the segment ST is 181 , which is approximately 13.45.
Examples
The distance formula is a fundamental concept in coordinate geometry and has numerous real-world applications. For example, civil engineers use it to calculate the lengths of roads or bridges on a map. Suppose two cities are located at coordinates A ( x 1 , y 1 ) and B ( x 2 , y 2 ) on a map. The engineer can use the distance formula to find the shortest distance between the two cities, which helps in planning efficient transportation routes. This ensures that resources and materials are used effectively, minimizing costs and environmental impact.
