If PQRS is a rectangle and M is the midpoint of RS, prove PM is congruent to QM.
P----------------Q
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R--------M-------S
*be as explanatory as possible
To prove that PM is congruent to QM, we need to show that the line segments PM and QM have the same length.
Given:
- PQRS is a rectangle.
- M is the midpoint of RS.
To prove that PM is congruent to QM, we can use the following properties of rectangles:
1. Opposite sides of a rectangle are congruent.
2. The diagonals of a rectangle are congruent and bisect each other.
Proof:
Since PQRS is a rectangle, we know that opposite sides are congruent. This means that PR is congruent to QS and PS is congruent to QR.
Since M is the midpoint of RS, it divides RS into two congruent segments. Thus, we can say that MS is congruent to MR.
Now, let's consider the triangles PMS and QMR.
In triangle PMS, we have:
- PM is a side of the triangle.
- MS and PS are the other two sides of the triangle.
In triangle QMR, we have:
- QM is a side of the triangle.
- MR and QR are the other two sides of the triangle.
From our earlier observations, we know that MS is congruent to MR (since M is the midpoint of RS), and PS is congruent to QR (since opposite sides of a rectangle are congruent).
Using the Side-Side-Side (SSS) congruence criterion:
- PM is congruent to QM (common side)
- MS is congruent to MR (given)
- PS is congruent to QR (given)
Therefore, by SSS congruence, we can conclude that triangle PMS is congruent to triangle QMR.
Since corresponding sides of congruent triangles are congruent, PM is congruent to QM.
Hence, we have proved that PM is congruent to QM.
To prove that PM and QM are congruent in rectangle PQRS, we can use the properties of a rectangle and the concept of midpoints.
First, let's start by establishing some properties of a rectangle. In a rectangle, opposite sides are parallel, and all angles are right angles (measuring 90 degrees).
Since PQRS is a rectangle, we know that PQ is parallel to RS, and RP is parallel to QS. Additionally, all four angles at the vertices P, Q, R, and S are right angles.
Let's focus on segment PM and QM. To prove that PM and QM are congruent, we need to show that they have the same length.
Since M is defined as the midpoint of RS, we can say that segment RM is congruent to segment MS. This is because in any line segment, the midpoint splits the segment into two equal halves. Therefore, RM is equal in length to MS.
Now, let's look at the triangle RPM and QMS. Both triangles share one side, RM or MS. We have established that RM is congruent to MS.
Additionally, PR is congruent to QS because they are opposite sides of the rectangle PQRS.
Lastly, since all angles in a rectangle are right angles, angle PRM and angle QSM are both right angles.
Using the Side-Angle-Side (SAS) congruence postulate, we can conclude that triangle RPM is congruent to triangle QMS.
Therefore, by the congruence of the triangles RPM and QMS, we can conclude that PM is congruent to QM.
In summary:
1. RM is congruent to MS because M is the midpoint of RS.
2. PR is congruent to QS because opposite sides of a rectangle are congruent.
3. Angle PRM and angle QSM are both right angles.
4. By the congruence of triangle RPM and triangle QMS, we can conclude that PM is congruent to QM.
Hence, we have proven that PM and QM are congruent in rectangle PQRS.
PRM and QSM are right triangles
RM=MS
PR=QS
by SAS, the triangles are congruent, so PM=QM