Evgeny.Makarov said:Any reflection across a line in the plane is an isometry (a distance-preserving transformation). In this special case it can be shown algebraically as follows. You know from another problem that the reflection of a point $(x,y)$ in the line $y=x$ has coordinates $(y,x)$. Let $P(x_1,y_1)$, $Q(x_2,y_2)$. Write the coordinates of $P'$ and $Q'$ and the distances $PQ$ and $P'Q'$.
Yes. The fact that $PQ=P'Q'$ can also be shown geometrically.RTCNTC said:Are you saying to use the distance formula for points?
Evgeny.Makarov said:Yes. The fact that $PQ=P'Q'$ can also be shown geometrically.
Evgeny.Makarov said:Suppose that $QQ'>PP'$. Drop perpendiculars $PR$ and $P'R'$ on $QQ'$. Let $M$ and $N$ be the midpoints of $PP'$ and $QQ'$, respectively. Then $PRNM$ and $P'R'NM$ are equal rectangles. Therefore $PR=P'R'$ and $QR=QN-RN=Q'N-R'N=Q'R'$. So $\triangle PQR=\triangle P'Q'R'$ and $PQ=P'Q'$.
Alternatively, Wikipedia says that for the trapezoid $PP'Q'Q$ to be isosceles it is sufficient that the segment $MN$ that joins the midpoints of the parallel sides is perpendicular to them, which is the case here by definition of symmetry.
Evgeny.Makarov said:Did you notice any problem with displaying LaTeX on other MHB pages? Try disabling any JavaScript blocker such as NoScript. If this does not help, the staff would appreciate if you submit a report in the http://mathhelpboards.com/questions-comments-feedback-25/ with a screenshot and browser version.
RTCNTC said:I do not care about LaTex.
MarkFL said:The vast majority of math helpers here, and indeed on every other math help site I know of, are going to use $\LaTeX$ when responding to questions. Reading anything but the simplest of expressions formatted in plain text is a chore at best.
So, if you care about being able to read the help with which you will be provided, it would be in your best interest to take steps to ensure you can read it. It may be as simple as using a better browser. :D
RTCNTC said:I do not have a computer or laptop. All my questions and replies are done via cell phone.
MarkFL said:Do you not have any way to try other browsers?
"Point P to Point Q" refers to a specific path or route between two points, P and Q. It is often used in mathematics and physics to describe the distance or trajectory between two points in space.
The distance between Point P and Point Q can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the distance between Point P and Point Q.
The path from Point P to Point Q can be affected by various factors such as obstacles, terrain, and curvature of the Earth's surface. These factors can alter the distance, direction, and difficulty of the path.
The shortest path from Point P to Point Q is determined using the concept of optimization. This involves finding the most efficient route that minimizes the distance or time required to travel between the two points.
Yes, the path from Point P to Point Q can change over time due to factors such as construction, natural disasters, or man-made changes to the environment. It is important to regularly recalculate the path in order to accurately determine the distance between the two points.