To prove that the diagonals of square PQRS are perpendicular bisectors of each other, we need to verify four key properties:
1. All four sides of the square are equal in length.
2. The slopes of the adjacent sides are perpendicular to each other.
3. Both diagonals are of equal length.
4. The slopes of the diagonals are perpendicular to each other.
Let's go through each property step-by-step:
### 1. Verify that all four sides of the square are equal in length.
The lengths of the sides are given as:
- [tex]\(\overline{SP} = 5\)[/tex]
- [tex]\(\overline{PQ} = 5\)[/tex]
- [tex]\(\overline{RQ} = 5\)[/tex]
- [tex]\(\overline{SR} = 5\)[/tex]
Since all these lengths are equal to 5, we confirm that the sides of the square are equal in length.
### 2. Verify the perpendicularity of the adjacent sides.
The slopes of the sides are given as:
- Slope of [tex]\(\overline{SP}\)[/tex] and [tex]\(\overline{RQ}\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex]
- Slope of [tex]\(\overline{SR}\)[/tex] and [tex]\(\overline{PQ}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex]
To confirm the perpendicularity, the product of the slopes of the adjacent sides should be [tex]\(-1\)[/tex]:
[tex]\[
\left(-\frac{4}{3}\right) \times \left(\frac{3}{4}\right) = -1
\][/tex]
As the product is [tex]\(-1\)[/tex], the adjacent sides are confirmed to be perpendicular.
### 3. Verify the equality of the diagonals’ lengths.
The lengths of the diagonals are given as:
- [tex]\(\overline{SQ} = \sqrt{50}\)[/tex]
- [tex]\(\overline{RP} = \sqrt{50}\)[/tex]
Since both values are equal, it confirms that the diagonals are of equal length.
### 4. Verify the perpendicularity of the diagonals.
The slopes of the diagonals are given as:
- Slope of [tex]\(\overline{RP}\)[/tex] is [tex]\(7\)[/tex]
- Slope of [tex]\(\overline{SQ}\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex]
To confirm the perpendicularity, the product of the slopes of the diagonals should be [tex]\(-1\)[/tex]:
[tex]\[
7 \times \left( -\frac{1}{7} \right) = -1
\][/tex]
As this condition is satisfied, the diagonals are perpendicular to each other.
### Conclusion
Since:
- All sides of the square are equal,
- The slopes of adjacent sides are perpendicular,
- The lengths of the diagonals are equal,
- The slopes of the diagonals are perpendicular,
We can conclude that the diagonals of square PQRS are indeed perpendicular bisectors of each other.
