Answer:
a) x = 7
b) x = 2
Step-by-step explanation:
* Lets revise some facts in the circle
- If two secant segments are drawn to a circle from a point outside the
circle, the product of the length of one secant segment and its
external part is equal to the product of the length of the other secant
segment and its external part.
# Example:
- If AC is a secant intersects the circle at points A and and B
- If DC is another secant intersects the circle at points D and E
- The two secants intersect each other out the circle at point C
∴ AC × CB = DC × CE , where AC is the secant and CB is its external
part and DC is the secant and CE is its external part
* Lets solve the problem
a) There are two secants intersect each other at point outside the circle
∵ The first secant is x + 5
∵ Its external part is 5
∵ the second secant is 4 + 6 = 10
∵ Its external part is 6
∴ (x + 5) × 5 = 10 × 6 ⇒ simplify
∴ 5x + 25 = 60 ⇒ subtract 25 from both sides
∴ 5x = 35 ⇒ divide both sides by 5
∴ x = 7
* x = 7
b) There are two secants intersect each other at point outside the circle
∵ The first secant is 5 + 3 = 8
∵ Its external part is 3
∵ the second secant is x + 4
∵ Its external part is 4
∴ 8 × 3 = (x + 4) × 4 ⇒ simplify
∴ 24 = 4x + 16 ⇒ subtract 16 from both sides
∴ 8 = 4x ⇒ divide both sides by 4
∴ 2 = x
* x = 2
