Subjects geometry

Distance Point Line

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Search Solutions

Distance Point Line


1. Stating the problem: Find the distance from the point $(2,3)$ to the straight line given by the equation $$4x + 3y = 10$$. 2. Formula for the distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is $$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$. 3. Rewrite the line equation into standard form $Ax + By + C = 0$: $$4x + 3y - 10 = 0$$ so here $A = 4$, $B = 3$, and $C = -10$. 4. Plug in $(x_0,y_0) = (2,3)$: $$|4(2) + 3(3) - 10| = |8 + 9 - 10| = |7| = 7$$. 5. Calculate denominator: $$\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$. 6. Calculate distance: $$d = \frac{7}{5} = 1.4$$. Answer: The distance from point $(2,3)$ to the line $4x + 3y = 10$ is $1.4$ units.