Question

Problem 11.8 - Enhanced - with Hints and Feedback 2016 The X and Y coordinates (in feet) of station Shore are 5048.64 and 3278.59, respectively, and those for station Rock are 3033.33 and 5876.93, respectively. Part A Suppose a point P is located near the straight line connecting stations Shore and Rock. What is the perpendicular distance from P to the line if the X and Y coordinates of point P are 4440.30 and 4254.14, respectively? Express your answer to three significant figures and include the appropriate units. View Available Hint(s) HHA O P ? Value Submit Previous Answers * Incorrect; Try Again; 4 attempts remaining

Answer 1

Hi,

Please see the solution of above question below,

Given Coordinates of ,

Point Shore (A) = (5048.64, 3278.59)

Point Rock (B) = (3033.33, 5876.93)

Point (P) = (4440.30, 4254.14)

Now,

Horizontal difference between shore and rock = 5048.64-3033.33 = 2015.31

Vertical difference between shore and rock = 5876.93-3278.59= 2598.34

Slope tan θ                         = -2598.34/2015.31               = -1.2893

Accordingly sin θ              =             -1.2893   .      = -0.7902

                                                                √1+(-1.2893)²

And cos θ                            =             √1-(-0.7902)²                                  = 0.6129

Now Putting the values of A in straight line equation y = mx + c,

3278.59 = -1.2893 x 5084.64 +c

c = 9787.802

Hence the equation of line AB is y = -1.2893 . x + 9787.802

for y = 0, x = 9787.802 -Ref. 1

Now lets assume a line CD parallel to line AB and intersect point P

Since both lines are parallel, slope (m) will be same for both lines.

So, equation of this line y_{1 =} mx₁ + c₁

putting the values of P & m

4440.30 = -1.2893 x 4254.14 + c₁

c₁ = 9979.019

Equation of line CD = y_{1 =} -1.2893 . x₁ + 9979.019

for y₁ = 0, x₁ = 9979.019 -Ref. 2

9979014 رم - مو 97870802/

From reference 1 & 2 vertical distance between AB & CD at Y- Y axis = 9979.019 - 9787.802 = 191.217 feet

Perpendicular distance between both lines = 191.217 x cos θ = 191.217 x 0.6129 = 117.197 feet

Hence the perpendicular distance from P to the line connecting shore and rock will be 117.197 feet.

Hope the same is clear now. In case of any query please leave comment.

Waiting to hear from you soon,

Warm Regards

Saras Jain