Geometry

To find the distance of the point from the line along the line parallel to , follow these steps:

1. Determine the direction vector of the parallel line:
The line can be rewritten in the slope-intercept form as . The direction vector is .

2. Perpendicular vector to the line :
This line can be rearranged as and further rewritten as . Its normal vector is .

3. Calculate the projection of the point onto this direction vector:
We find the dot product of and :

The magnitude of the direction vector is .

4. Distance from a point to a line formula:
The distance from the point to the line is given by:

Substitute the point into the formula:

=

5. Adjusting the distance along the direction vector:
The actual distance along the line parallel to is scaled by , as derived from the division of magnitudes. Therefore, the adjusted distance is:

Returning the solution gives:

Hence, the distance is .

A polygon with sides can have diagonals that are calculated using the formula:

We know the polygon has 105 diagonals. Thus, we set up the equation:

Solve for by first multiplying both sides by 2 to clear the fraction:

Now, expand and rearrange to form a standard quadratic equation:

Next, we factorize the quadratic equation:

The solutions for are obtained by setting each factor to zero:

• leads to
• leads to

We disregard as a polygon cannot have negative sides.

Therefore, the solution is:

The polygon has 15 sides.

Midpoint Calculation and Median Equation

We are given the following information:

• The coordinates of point are .
• The equation of the median through is .
• The equation of the median through is .

Step 1: Find the midpoint of

The median through is the line , meaning that point lies on the vertical line where . Let the coordinates of be .

Step 2: Midpoint formula

The midpoint of a line segment joining two points and is given by the formula:

We know that the midpoint lies on the median through , and the coordinates of are . Therefore, the midpoint of also lies on the line joining and the midpoint of , with coordinates .

Step 3: Substitute the value for the midpoint

Now we can substitute the value of the midpoint into the equation of the median through and solve for and using the given equations.

We know that the median through has the equation . Since the midpoint lies on this line, substitute the coordinates of into the equation:

Multiply through by 2 to simplify:

Now, simplify the equation:

Thus, the sum of the coordinates of point is . From this, you can solve for the coordinates of based on the known values of , or use further algebraic manipulation to get the precise coordinates of and .