The point Q is the image of the point P(1,5) about the line y=x and R is the image of the point Q about the line `y= -x`. The circumcentre of the `DeltaPQR` is

Options

A. (5,1)
B. `(-5,1)`
C. `(1,-5)`
D. (0,0)

Answer

To find the circumcenter of triangle PQR, we will follow these steps:

### Step 1: Find the coordinates of point Q, the image of point P(1, 5) about the line y = x.

The line y = x is a line of symmetry. To find the image of point P(1, 5) about this line, we can use the following formula for the reflection of a point (x1, y1) about the line y = x:

- The image point Q will have coordinates (y1, x1).

Thus, for point P(1, 5):
- Q = (5, 1)

### Step 2: Find the coordinates of point R, the image of point Q(5, 1) about the line y = -x.

To find the image of point Q(5, 1) about the line y = -x, we can use the formula for reflection about this line:

- The image point R will have coordinates (-y1, -x1).

Thus, for point Q(5, 1):
- R = (-1, -5)

### Step 3: Identify the coordinates of points P, Q, and R.

Now we have the coordinates of all three points:
- P = (1, 5)
- Q = (5, 1)
- R = (-1, -5)

### Step 4: Find the midpoints of segments PQ and QR.

To find the circumcenter of triangle PQR, we need to find the midpoints of at least two sides of the triangle.

**Midpoint of PQ:**
- Midpoint M1 = ((x1 + x2)/2, (y1 + y2)/2)
- M1 = ((1 + 5)/2, (5 + 1)/2) = (3, 3)

**Midpoint of QR:**
- Midpoint M2 = ((5 + (-1))/2, (1 + (-5))/2) = (2, -2)

### Step 5: Find the slopes of PQ and QR.

**Slope of PQ (mPQ):**
- mPQ = (y2 - y1) / (x2 - x1) = (1 - 5) / (5 - 1) = -4 / 4 = -1

**Slope of QR (mQR):**
- mQR = (-5 - 1) / (-1 - 5) = -6 / -6 = 1

### Step 6: Find the equations of the perpendicular bisectors of PQ and QR.

**Perpendicular bisector of PQ:**
- The slope of the perpendicular bisector is the negative reciprocal of mPQ, which is 1.
- Using point M1(3, 3), we can write the equation:
- y - 3 = 1(x - 3) → y = x

**Perpendicular bisector of QR:**
- The slope of the perpendicular bisector is the negative reciprocal of mQR, which is -1.
- Using point M2(2, -2), we can write the equation:
- y + 2 = -1(x - 2) → y = -x

### Step 7: Find the intersection of the two perpendicular bisectors.

To find the circumcenter, we need to solve the equations:
1. y = x
2. y = -x

Setting these equal to each other:
- x = -x → 2x = 0 → x = 0
- Substitute x = 0 into y = x → y = 0

Thus, the circumcenter of triangle PQR is at the point (0, 0).

### Final Answer:
The circumcenter of triangle PQR is (0, 0).

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