Final answer:
To find the length of line segment MN on the Cartesian plane, we use the Pythagorean theorem. The difference in coordinates gives us two sides of a right-angled triangle where MN is the hypotenuse. The calculation results in MN being 2√2 units long.
Step-by-step explanation:
The student is asking to find the length of line segment MN on a Cartesian plane, where M and N are points with given coordinates. To find the length between two points, which is also known as the distance formula, we can utilize the Pythagorean theorem in the context of a two-dimensional Cartesian coordinate system. By calculating the differences in the x-coordinates and y-coordinates separately, we form a right-angled triangle where the line segment MN is the hypotenuse.
Given M(10,4) and N(12,6), the difference in x-coordinates (Δx) is 12 - 10 = 2, and the difference in y-coordinates (Δy) is 6 - 4 = 2. Applying the distance formula:
Length MN = √((Δx)² + (Δy)²)
Length MN = √(2² + 2²)
Length MN = √(4 + 4)
Length MN = √8
Since the question asks for the answer in surds form, we express the square root of 8 as the square root of 4 times 2, which simplifies to 2 times the square root of 2.
Length MN = 2√2 units