In the xy-plane, line k has a slope of \frac{3}{4} and passes through the point (2, 1). Li… — GRE General Test

In the $xy$ -plane, line $k$ has a slope of $\frac{3}{4}$ and passes through the point $(2, 1)$ . Line $k$ intersects the line $x = 10$ at point $P$ . What is the distance between point $P$ and the point $(2, 1)$ ?

Answer: 10

Answer

Point P lies on the line x = 10, so its x-coordinate is 10. The horizontal change from x = 2 to x = 10 is 8 units. Given that line k has slope 3/4, the corresponding vertical change is (3/4)(8) = 6 units. Therefore, point P has coordinates (10, 7). The distance between (2, 1) and (10, 7) is sqrt((10 - 2)^2 + (7 - 1)^2) = sqrt(64 + 36) = sqrt(100) = 10.

Step-by-Step Solution

Determine the coordinates of point P

P is located at (10, 7)

Since P lies on the line x = 10, its x-coordinate is 10. The horizontal distance from (2, 1) to P is 10 - 2 = 8. Using the slope m = 3/4, the vertical change is (3/4) * 8 = 6, so the y-coordinate of P is 1 + 6 = 7.

Calculate the distance between (2, 1) and (10, 7)

The distance is 10

Using the distance formula sqrt((10 - 2)^2 + (7 - 1)^2) = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10.

Key Concept

Slope definition and distance formula in coordinate geometry

Estimated Time:1m 30s

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