If PQ=5x+2 and QS=12x-23, and QR=x+5 and PQ is congruent to RA, then what does x and RS equal?

Answer 1

Are P, Q, R, and S points on the same line? Also, is "RA" meant to be "RS"? This is just a guess because A and S are next to each other on the keyboard, and the problem doesn't otherwise mention a point A. If both of these assumptions are correct, the problem can be solved as follows:

First, it will help you to draw a straight line which connects the points P, Q, R, and S. This will help you visualize the problem.

Then, the problem gives the length of line segment QS and of line segment QR in terms of x. You can subtract as follows to get the length of line segment RS in terms of x, where all three values are the lengths of line segments:

RS = QS - QR

Now, consider that if two shapes are congruent, they have the same size. For line segments, this means having the same length.

Therefore, we can write this equation, where both values again represent the lengths of line segments:

PQ = RS

In this equation, you can replace PQ with the given value in terms of x. You can then replace RS with the expression in terms of x which you determined above. Then, you can use algebra to solve for x.

Finally, you can plug in the value of x into the expression for RS to determine the length of line segment RS as an integer.

Answer 2

Assuming you mean RS and not RA, and also assuming that the points are colinear and in alphabetical order (P-Q-R-S)...

From overlapping segments, you get that RS = QS - QR. This means that RS = (12x-23) - (x+5), or 11x-28.

Since RS is congruent with PQ, this means that their lengths are equal: 5x+2 = 11x-28. From here, just solve for x:

5x+2 = 11x-28

==> -6x+2 = -28

==> -6x = -30

==> x=5

So x equals 5, and RS has a length of 27.