Point P (2,-7) is the centre of a circle with radius 13 units, PT is perpendicular to chord AB and T= (-2,-4). Calculate the length
AT

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the Given Information
- Center of the circle, \( P(2, -7) \)
- Radius of the circle, \( r = 13 \) units
- Point \( T(-2, -4) \) where \( PT \) is perpendicular to chord \( AB \)

### Step 2: Calculate the Length of \( PT \)
We will use the distance formula to find the length of \( PT \). The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points \( P(2, -7) \) and \( T(-2, -4) \):

\[ PT = \sqrt{((-2) - 2)^2 + ((-4) - (-7))^2} \]
\[ PT = \sqrt{(-4)^2 + (3)^2} \]
\[ PT = \sqrt{16 + 9} \]
\[ PT = \sqrt{25} = 5 \]

### Step 3: Use the Pythagorean Theorem in Triangle \( APT \)
Since \( PT \) is perpendicular to chord \( AB \), triangle \( APT \) is a right triangle. We can apply the Pythagorean theorem:

\[ AP^2 = PT^2 + AT^2 \]

Where:
- \( AP \) is the radius of the circle, which is 13 units.
- \( PT \) is calculated as 5 units.
- \( AT \) is the length we want to find.

Substituting the known values:

\[ 13^2 = 5^2 + AT^2 \]
\[ 169 = 25 + AT^2 \]
\[ AT^2 = 169 - 25 \]
\[ AT^2 = 144 \]

### Step 4: Calculate \( AT \)
Taking the square root of both sides:

\[ AT = \sqrt{144} = 12 \]

### Final Answer
The length of \( AT \) is **12 units**.

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