If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is

Given:

Radii of two intersecting circles: r₁ = 5 cm and r₂ = 12 cm

The angle of intersection at the point where the two circles meet is 90°.

We are required to find the length of the common chord formed by the intersection of the two circles.

Step 1: Use the formula for the length of the common chord

The length of the common chord L for two intersecting circles with radii r₁ and r₂ at an angle θ = 90° between them can be found using the following formula:

L = 2√[r₁² - (d² + r₁² - r₂²) / (2d)]²

Where:

d is the distance between the centers of the two circles.

Step 2: Understanding the Geometry

When the two circles intersect at a 90-degree angle:

The distance between the centers of the two circles (denoted as d) can be calculated using the Pythagorean theorem as:

d = √(r₁² + r₂²)

Step 3: Calculate the distance between the centers

Substitute the known values of the radii:

d = √(5² + 12²) = √(25 + 144) = √169 = 13 cm

Step 4: Apply the formula for the common chord length

Now that we know the distance between the centers d = 13 cm, we can calculate the length of the common chord using the formula:

L = 2√[r₁² - (d² + r₁² - r₂²) / (2d)]²

Substitute the known values:

L = 2√[5² - ((13² + 5² - 12²) / (2 × 13))]²

Step 5: Simplify the expression

(13² + 5² - 12²) / (2 × 13) = (169 + 25 - 144) / 26 = 50 / 26 = 25 / 13

Step 6: Calculate the common chord length

Now, calculate L:

L = 2√[25 - (25 / 13)²] 

L = 2√[25 - (625 / 169)] 

L = 2√[(25 × 169) / 169 - 625 / 169] 

L = 2√[4225 / 169 - 625 / 169] 

L = 2√[3600 / 169] 

L = 2 × (60 / 13) 

L = 120 / 13

Final Answer

The length of the common chord is 120 / 13 cm.