Answer:
DC = 6
Explanation:
Given in the question,
radius = 3x
tangents to the circle = 12 + 5x and 8 + 7x
If you draw line from A to D, two right angle triangle will be form ADC and ADB
having radius = 3x
By using pythagorus theorem
hypotenuse² = base² + height²
Since, both triangle share same hypotenuse AD
so,
ΔADC ΔADB
base² + height² = base² + height²
3x² + (8 + 7x)² = 3x² + (12 + 5x)²
3x² will be cancel out
(8 + 7x)² = (12 + 5x)²
64 + 112x + 49x² = 144 + 120x + 25x²
rearranging like terms
49x²-25x² +112x - 120x + 64 -144 = 0
24x² - 8x - 80 = 0
(x- 2)(x + 5/3)
x = 2
x = -5/3 rejected as radius can't be negative
So,
length of radius = 3x = 6