Question

solve the problem using inductive reasoning. how many line segments are determined by joining dots on the last two circles? 3 segments 6 segments? segments? segments 11,29 12,30 10,28 9,27

Explanation:

Step1: Identify dot count per circle

1st circle: 3 dots; 2nd: 4 dots; 3rd: 6 dots; 4th: 8 dots

Step2: Use combination formula

The number of line segments formed by $n$ non-collinear points is given by the combination formula $\binom{n}{2} = \frac{n(n-1)}{2}$, which counts unique pairs of points (each pair forms one segment).

Step3: Calculate for 3rd circle (6 dots)

Substitute $n=6$ into the formula:
$\frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15$
Wait, correction: Recheck the pattern with first two circles:

• 3 dots: $\frac{3(3-1)}{2}=3$ (matches given)
• 4 dots: $\frac{4(4-1)}{2}=6$ (matches given)
• 6 dots: $\frac{6(6-1)}{2}=15$
• 8 dots: $\frac{8(8-1)}{2}=28$

Wait, no—wait the options have 10,28; recheck the 3rd circle dot count: the 3rd circle has 5 dots (count the dots: <point>370 300</point><point>370 395</point><point>410 420</point><point>439 300</point><point>439 395</point> → 5 dots)

Step1 (corrected): Count dots accurately

1st: 3 dots; 2nd:4 dots; 3rd:5 dots; 4th:8 dots

Step2: Apply combination formula to 3rd circle (5 dots)

$\frac{5(5-1)}{2} = \frac{5 \times 4}{2} = 10$

Step3: Apply combination formula to 4th circle (8 dots)

$\frac{8(8-1)}{2} = \frac{8 \times 7}{2} = 28$

Answer:

10,28