There are two chords AB and AC of equal length 8 cm . CB is produced to P. AP cuts circle at T s.t. AT = 5 cm . find PT

Answer

To solve the problem, we will follow these steps:

### Step-by-Step Solution:

1. **Draw the Circle and Chords**:
- Start by drawing a circle. Mark two points A and B on the circumference of the circle such that the length of chord AB is 8 cm.
- Now, mark point C on the circumference such that the length of chord AC is also 8 cm.

2. **Extend CB to Point P**:
- Extend the line segment CB beyond point C to a new point P.

3. **Identify Point T**:
- Draw a line from point A that intersects the circle again at point T. We know that AT = 5 cm.

4. **Label the Lengths**:
- Since AB = AC = 8 cm, we have AB = 8 cm and AC = 8 cm.
- Let the length of segment PT be denoted as x. Therefore, the length of segment PA will be AT + PT = 5 cm + x.

5. **Use Similar Triangles**:
- We can use the properties of similar triangles here. The triangles PAB and TAB are similar because they share angle A and have equal angles at B and T (since AB = AC).
- By the property of similar triangles, we can set up a proportion:
\[ \frac{PA}{AB} = \frac{AT}{PT} \]

6. **Substitute the Known Values**:
- Substitute the known lengths into the proportion:
\[ \frac{5 + x}{8} = \frac{5}{x} \]

7. **Cross Multiply**:
- Cross multiplying gives us:
\[ (5 + x) \cdot x = 5 \cdot 8 \]
\[ 5x + x^2 = 40 \]

8. **Rearrange the Equation**:
- Rearranging the equation gives us:
\[ x^2 + 5x - 40 = 0 \]

9. **Solve the Quadratic Equation**:
- We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 5\), and \(c = -40\).
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-40)}}{2 \cdot 1} \]
\[ x = \frac{-5 \pm \sqrt{25 + 160}}{2} \]
\[ x = \frac{-5 \pm \sqrt{185}}{2} \]
\[ x = \frac{-5 \pm 13.6}{2} \]

10. **Calculate the Positive Root**:
- We take the positive root since length cannot be negative:
\[ x = \frac{8.6}{2} = 4.3 \text{ cm} \]

11. **Final Length of PT**:
- Therefore, the length of PT is:
\[ PT = x = 4.3 \text{ cm} \]

### Final Answer:
PT = 4.3 cm