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Sagot :
To prove that the diagonals of square PQRS are perpendicular bisectors of each other, let's analyze the given information step by step.
### Step-by-Step Solution
1. Lengths of the Sides:
- We know the lengths of all sides of the square PQRS are equal to 5 units. Hence,
[tex]\[ \overline{SP} = \overline{PQ} = \overline{RQ} = \overline{SR} = 5 \][/tex]
2. Slopes of the Sides:
- The slope of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is:
[tex]\[ -\frac{4}{3} \][/tex]
- The slope of [tex]\(\overline{SR}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is:
[tex]\[ \frac{3}{4} \][/tex]
3. Lengths of the Diagonals:
- The lengths of both diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are:
[tex]\[ \sqrt{50} \][/tex]
4. Midpoint of the Diagonals:
- Both diagonals have a common midpoint given by:
[tex]\[ \left( 4 \frac{1}{2}, 5 \frac{1}{2} \right) \][/tex]
5. Slopes of the Diagonals:
- The slope of the diagonal [tex]\(\overline{RP}\)[/tex] is:
[tex]\[ 7 \][/tex]
- The slope of the diagonal [tex]\(\overline{SQ}\)[/tex] is:
[tex]\[ -\frac{1}{7} \][/tex]
6. Proof of Perpendicularity:
- For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. Let's check the product of the slopes of the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex]:
[tex]\[ \text{slope}_{\overline{RP}} \times \text{slope}_{\overline{SQ}} = 7 \times -\frac{1}{7} = -1 \][/tex]
- Therefore, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular.
7. Proof of Bisection:
- Since both diagonals have the same lengths ([tex]\(\sqrt{50}\)[/tex]) and share a common midpoint [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], it means they bisect each other at this point. This is because the midpoint is equidistant from the endpoints of both diagonals.
### Conclusion
- The diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular because their slopes multiply to [tex]\(-1\)[/tex].
- The diagonals bisect each other at [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], as they share a common midpoint.
- Therefore, the diagonals of square PQRS are perpendicular bisectors of each other, thus confirming the properties of the diagonals in a square.
Hence, all the conditions are satisfied, proving that the diagonals of square PQRS are perpendicular bisectors of each other.
### Step-by-Step Solution
1. Lengths of the Sides:
- We know the lengths of all sides of the square PQRS are equal to 5 units. Hence,
[tex]\[ \overline{SP} = \overline{PQ} = \overline{RQ} = \overline{SR} = 5 \][/tex]
2. Slopes of the Sides:
- The slope of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is:
[tex]\[ -\frac{4}{3} \][/tex]
- The slope of [tex]\(\overline{SR}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is:
[tex]\[ \frac{3}{4} \][/tex]
3. Lengths of the Diagonals:
- The lengths of both diagonals [tex]\(\overline{SQ}\)[/tex] and [tex]\(\overline{RP}\)[/tex] are:
[tex]\[ \sqrt{50} \][/tex]
4. Midpoint of the Diagonals:
- Both diagonals have a common midpoint given by:
[tex]\[ \left( 4 \frac{1}{2}, 5 \frac{1}{2} \right) \][/tex]
5. Slopes of the Diagonals:
- The slope of the diagonal [tex]\(\overline{RP}\)[/tex] is:
[tex]\[ 7 \][/tex]
- The slope of the diagonal [tex]\(\overline{SQ}\)[/tex] is:
[tex]\[ -\frac{1}{7} \][/tex]
6. Proof of Perpendicularity:
- For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. Let's check the product of the slopes of the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex]:
[tex]\[ \text{slope}_{\overline{RP}} \times \text{slope}_{\overline{SQ}} = 7 \times -\frac{1}{7} = -1 \][/tex]
- Therefore, the diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular.
7. Proof of Bisection:
- Since both diagonals have the same lengths ([tex]\(\sqrt{50}\)[/tex]) and share a common midpoint [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], it means they bisect each other at this point. This is because the midpoint is equidistant from the endpoints of both diagonals.
### Conclusion
- The diagonals [tex]\(\overline{RP}\)[/tex] and [tex]\(\overline{SQ}\)[/tex] are perpendicular because their slopes multiply to [tex]\(-1\)[/tex].
- The diagonals bisect each other at [tex]\(\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)\)[/tex], as they share a common midpoint.
- Therefore, the diagonals of square PQRS are perpendicular bisectors of each other, thus confirming the properties of the diagonals in a square.
Hence, all the conditions are satisfied, proving that the diagonals of square PQRS are perpendicular bisectors of each other.
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