Line ℓ has equation y=x-3. Find the distance between ℓ and the point E(-1,3). Round your answer to the nearest tenth.

Question

Line  ℓ has equation  y=x-3. Find the distance between  ℓ and the point  E(-1,3). Round your answer to the nearest tenth.

Bytelearn · Accepted Answer

Identify Slope and Point: Identify the slope and a point on line ℓ. The equation of line ℓ is y = x - 3, which is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. The slope of line ℓ is 1 (since the coefficient of x is 1), and the y-intercept is -3. A point on line ℓ can be found by plugging in any value for x. For simplicity, we can use the y-intercept point (0, -3).Find Perpendicular Line Equation: Find the equation of the line perpendicular to ℓ that passes through point E(-1, 3). The slope of a line perpendicular to ℓ will be the negative reciprocal of the slope of ℓ. Since the slope of ℓ is 1, the slope of the perpendicular line will be -1. Using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we can plug in the slope and the coordinates of point E to find the equation of the perpendicular line. y - 3 = -1(x - (-1)) y - 3 = -1(x + 1) y - 3 = -x - 1 y = -x + 2Find Point of Intersection: Find the point of intersection between line ℓ and the perpendicular line. To find the point of intersection, we set the equations of the two lines equal to each other and solve for x. x - 3 = -x + 2 2x = 5 x = 5/2 Now, plug x back into the equation of line ℓ to find the corresponding y-coordinate. y = (5/2) - 3 y = 5/2 - 6/2 y = -1/2 The point of intersection is (5/2, -1/2).Calculate Distance: Calculate the distance between point E and the point of intersection. The distance between two points (x1, y1) and (x2, y2) is given by the formula: Distance = √((x2 - x1)² + (y2 - y1)²) Plugging in the coordinates of point E (-1, 3) and the point of intersection (5/2, -1/2), we get: Distance = √((5/2 - (-1))² + (-1/2 - 3)²) Distance = √((5/2 + 1)² + (-1/2 - 3)²) Distance = √((5/2 + 2/2)² + (-1/2 - 6/2)²) Distance = √((7/2)² + (-7/2)²) Distance = √(49/4 + 49/4) Distance = √(98/4) Distance = √(49/2) Distance = √(24.5) Distance ≈ 4.95 Round the answer to the nearest tenth: 5.0

Line $\ell$ has equation $y=x-3$ . Find the distance between $\ell$ and the point $E(-1,3)$ . $\newline$ Round your answer to the nearest tenth.

Full solution

More problems from Write an equation for a parallel or perpendicular line

Related topics

Line ℓ \ell ℓ has equation y=x−3 y=x-3 y=x−3. Find the distance between ℓ \ell ℓ and the point E(−1,3) E(-1,3) E(−1,3).\newlineRound your answer to the nearest tenth.

Line $\ell$ has equation $y=x-3$ . Find the distance between $\ell$ and the point $E(-1,3)$ . $\newline$ Round your answer to the nearest tenth.