# If PQ=RQ, then which of the following must be true ?

(1) Q is the midpoint of PR (2) Q is on the perpendicular bisector of PR (3) PQ +QR = PR (4) Q is between P and R.

• Anonymous
1 month ago

There are two definitions of a perpendicular bisector:

(i) A line that passes through the midpoint and is perpendicular to the line

(ii) A line that contains all points that are equidistant from the two endpoints of the line.

Using definition (ii), the second statement is true.

I hope this helps!

• Raymond
1 month ago

Since you do not state that R is distinct from P, then it is possible that point Q is exactly at the same place as point P.

In such a case, then, all 4 statements could be false.

Let the point P be at the origin

P(0, 0)

Let the point Q be one unit to the right on the x-axis

Q(1, 0)

Let the point R be at the origin

R(0, 0)

[there is no rule that forbids two points to be at the same spot, unless you state that the points must be distinct]

Then the distance PQ = 1.

The distance QR is also 1.

PQ = RQ

1) Q is not the midpoint of PR (the mid point of PR would be (0, 0))

2) The perpendicular of PR is undefined.

3) PQ + QR = 1 + 1 (distances are never negative); while PR = 0

HOWEVER, if you mean PQ and QR as vectors, then the vector QR is (-1 0) while PQ = (+1 0) and, in such as case, statement number 3 would be true.

4) Q(1, 0) is not between P(0, 0) and R(0, 0)

• G
1 month ago

I am guessing more than one can be true, because the way you have written this question, that is the case.

If Q is the midpoint of PR, then PQ must be the same length as QR. So 1, 3, and 4 are true.

• Anonymous
1 month ago

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. . . . . . . . . . . . . . . Q. . . . . . . . . . . . . . . .

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P ————————–M——————————R

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This is the drawing most suitable for your problem. M is the middle of PR. MQ is the perpendicular bisector of PR.

The correct answer is (2) because any point on the perpendicular bisector is at the same distance from P as from R. And because of that we have that the answers (1), (3) and (4) are not true for any point Q that respects your assumption that PQ = QR.

PS: sorry for the incomplete drawing, but it’s not my fault… PR is longer; R is on the right side.

• Iggy Rocko
1 month ago