ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal. Show that: (i) SR || AC and SR = (1/2)AC (ii) PQ = SR (iii) PQRS is... ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal. Show that: (i) SR || AC and SR = (1/2)AC (ii) PQ = SR (iii) PQRS is a parallelogram.
Understand the Problem
The problem describes a quadrilateral ABCD where P, Q, R, and S are the midpoints of sides AB, BC, CD, and DA respectively. AC is a diagonal. We need to prove the following: (i) SR is parallel to AC and SR is half of AC. (ii) PQ = SR (iii) PQRS is a parallelogram.
Answer
(i) $SR \parallel AC$ and $SR = \frac{1}{2}AC$ (ii) $PQ = SR$ (iii) $PQRS$ is a parallelogram
Answer for screen readers
(i) $SR \parallel AC$ and $SR = \frac{1}{2}AC$ (ii) $PQ = SR$ (iii) $PQRS$ is a parallelogram
Steps to Solve
- Consider triangle ADC
$S$ and $R$ are midpoints of $AD$ and $DC$ respectively. By the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of it.
- Apply the midpoint theorem
Applying the midpoint theorem to triangle $ADC$, we have $SR \parallel AC$ and $SR = \frac{1}{2}AC$. So, we've proven part (i).
- Consider triangle ABC
$P$ and $Q$ are midpoints of $AB$ and $BC$ respectively. Again, by the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of it.
- Apply the midpoint theorem again
Applying the midpoint theorem to triangle $ABC$, we have $PQ \parallel AC$ and $PQ = \frac{1}{2}AC$.
- Compare PQ and SR
Since $SR = \frac{1}{2}AC$ and $PQ = \frac{1}{2}AC$, we can conclude that $PQ = SR$. This proves part (ii).
- Prove PQRS is a parallelogram
We know $SR \parallel AC$ and $PQ \parallel AC$, therefore $SR \parallel PQ$. Also, we already know that $PQ = SR$. Since one pair of opposite sides ($PQ$ and $SR$) of quadrilateral $PQRS$ is equal and parallel, $PQRS$ is a parallelogram. This proves part (iii).
(i) $SR \parallel AC$ and $SR = \frac{1}{2}AC$ (ii) $PQ = SR$ (iii) $PQRS$ is a parallelogram
More Information
The midpoint theorem is a fundamental concept in geometry that simplifies proofs involving midpoints of triangle sides.
Tips
A common mistake is not recognizing the applicability of the midpoint theorem. Another mistake is incorrectly applying the midpoint theorem, for example, stating $SR = AC$ instead of $SR = \frac{1}{2}AC$.
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