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In the rectangle OABC, M is the midpoint of OA and N is the midpoint of AB. OB meets MC at P and NC at Q. Show that OP=PQ=QB
google is your friend. You might start here:
https://math.stackexchange.com/questions/1148100/proving-that-lines-from-any-vertex-of-a-parallelogram-to-the-midpoints-of-the-op
To show that OP = PQ = QB, we need to prove that the lengths of these segments are equal in the given rectangle OABC.
Let's break down the problem step by step:
Step 1: Draw the rectangle OABC and mark the midpoints:
Start by drawing a rectangle with vertices O, A, B, and C. Then, mark the midpoint of OA as M and the midpoint of AB as N.
Step 2: Draw the diagonals:
Draw the diagonal OB, dividing the rectangle into two triangles, OBM and OBC. Also, draw the diagonal NC, dividing the rectangle into two triangles, ANC and BNC.
Step 3: Prove that OP = PQ:
To prove that OP = PQ, we need to show that the lengths of these segments are equal.
To do this, we can use the Midpoint Theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Since M is the midpoint of OA and N is the midpoint of AB, we can apply the Midpoint Theorem to triangle OAB. From this, we conclude that MN is parallel to OB and half its length.
Since MN is parallel to OB, we can apply the Parallel Lines Theorem. According to this theorem, corresponding angles formed by a transversal (in this case, MN) crossing two parallel lines (in this case, OB and MC) are congruent.
Using the Parallel Lines Theorem, we can conclude that angle MON = angle OBP.
Now, since we have a pair of congruent angles and a pair of corresponding sides proportional (MN is parallel to OB), we can conclude that triangles MON and OBP are similar.
Since these triangles are similar, we can use the Side-Splitter Theorem, which states that if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.
From the Side-Splitter Theorem, we know that MO/OM = BP/PQ, and since M is the midpoint of OA, MO = OM. Therefore, BP = PQ.
Hence, we have shown that OP = PQ.
Step 4: Prove that PQ = QB:
To prove that PQ = QB, we need to show that the lengths of these segments are equal.
Since P is the intersection point of OB and MC, we can use the properties of parallel lines to demonstrate this.
First, notice that MN is parallel to BC (as MN is parallel to OB, and BC is also parallel to OB as opposite sides of a rectangle are parallel).
Using the same reasoning as in Step 3, you can show that triangles MON and CBN are similar.
Applying the Side-Splitter Theorem once again, we can conclude that NC/CB = QP/PB, and since NC = CB (as they are opposite sides of a rectangle), we have QP = PQ = QB.
Hence, we have shown that OP = PQ = QB, as desired.
Overall, by using the properties of parallel lines, the Midpoint Theorem, the Parallel Lines Theorem, the Side-Splitter Theorem, and the properties of rectangles, we have shown that OP = PQ = QB in the rectangle OABC.