To prove that \( PQ = PR \) given that \( PS \) is a median of \( \triangle PQR \) and \( PS = QR \), we will use properties of triangles and medians.
Given:
- \( PS \) is a median of \( \triangle PQR \). By the definition of a median, it divides the opposite side (here \( QR \)) into two equal halves.
- \( PS = QR \).
To prove:
- \( PQ = PR \).
### Proof:
Since \( PS \) is a median of \( \triangle PQR \):
1. **Property of Medians:**
- In a triangle, a median divides the opposite side into two equal halves.
2. **Given \( PS = QR \):**
- This implies that \( PS \) is equal to the entire side \( QR \).
Now, let's use these properties to prove \( PQ = PR \):
- Consider \( \triangle PQR \) with \( PS \) as the median.
- Since \( PS \) is a median, it divides \( QR \) into two equal halves.
- Given \( PS = QR \), we have \( PS \) equal to the entire side \( QR \).
By the property of medians, \( PS \) divides \( QR \) into two segments \( QS \) and \( SR \) such that \( QS = SR \).
Therefore, \( PQ \) and \( PR \) are the sums of \( QS \) and \( PS \), and
[tex] \huge \bf \blue{-: ANSWER :-}[/tex]
We are given that triangle PQR has PS as a median (which means PS connects P to the midpoint of QR) and PS = QR. We need to prove that PQ = PR (sides adjacent to the same base).
Here's how we can prove it:
1. Since PS is a median: This implies that point S is the midpoint of QR. In other words, QS = SR.
2. Given information:We are also given that PS = QR.
3. Triangle side properties: In any triangle, the sum of the lengths of two sides must be greater than the length of the third side (Triangle Inequality Theorem).
Now, let's consider the sides of triangles PQS and PRS:
Triangle PQS: PQ + QS > PS (from Triangle Inequality)
Triangle PRS:PR + SR > PS (from Triangle Inequality)
4. Substitute the given information:
In the inequality for triangle PQS, substitute QS = SR (from point 1) and PS = QR (from point 2): PQ + SR > QR
In the inequality for triangle PRS, substitute SR = QS (from point 1) and PS = QR (from point 2): PR + QS > QR
5. Combine the inequalities: Since both PQ + SR and PR + QS are greater than QR, we can conclude that PQ > QR - SR and PR > QR - QS.
However, we know from point 1 that SR = QS. Therefore, both PQ > QR - SR and PR > QR - SR become:
PQ > QR - SR (which becomes PQ > QR - QS since SR = QS)
PR > QR - SR (which becomes PR > QR - QS since SR = QS)
Since these inequalities hold true, it's impossible for both PQ and PR to be strictly larger than QR - QS (which is the same for both sides). This leaves us with two possibilities:
Case 1: PQ = QR - SR (and PR = QR - SR).
If this were the case, then PQ and PR would both be equal to half of QR, which contradicts the given information that PS (which is the same length as QR) is a median. A median divides the base (QR in this case) into two segments with equal lengths.
Case 2:PQ = PR (and both must be less than or equal to QR - SR).
This is the only scenario that satisfies the given information and the Triangle Inequality Theorem.
Therefore, we can conclude that PQ must be equal to PR for the given conditions to hold true.
[Proved]
