the graphs below shows some properties of regular polygons.\nwhen compared with the independent variable…

Explanation:

Step1: Recall linear relationship definition

A linear relationship between two variables (x) (independent) and (y) (dependent) has a constant rate of change, meaning the graph is a straight line (either increasing, decreasing, or horizontal with constant slope).

Step2: Analyze the first graph (Number of Sides vs # of Diagonals)

The points are at ((3,0)), ((4,2)), ((5,5)), ((8,20)). Let's check the slope between consecutive points:

• From ((3,0)) to ((4,2)): slope (m_1=\frac{2 - 0}{4 - 3}=2)
• From ((4,2)) to ((5,5)): slope (m_2=\frac{5 - 2}{5 - 4}=3)
• From ((5,5)) to ((8,20)): slope (m_3=\frac{20 - 5}{8 - 5}=5) The slopes are not constant, so this is not linear.

Step3: Analyze the second graph (assuming it's related to interior angles or something else, but the visible point is at the end with a horizontal line? Wait, the second graph has a point at the end with (y = 720), but if it's a horizontal line (constant (y) as (x) changes), the slope is 0 (constant). Wait, but maybe the second graph: if the independent variable is number of sides, for a regular polygon, the sum of interior angles is ((n - 2)\times180). Wait, no, the second graph's (y)-axis is 720? Wait, maybe the second graph: if it's a horizontal line (constant (y) regardless of (x)), then the slope is 0 (constant), so it's linear (horizontal line, constant rate of change (0)). Wait, but the first graph is not linear. Wait, maybe I misread. Wait, the first graph: let's re - check. Wait, the first graph: number of sides ((n)) and number of diagonals ((d)) have the formula (d=\frac{n(n - 3)}{2}), which is a quadratic function (not linear). The second graph: if it's a horizontal line (e.g., maybe the sum of exterior angles, which is always (360^\circ) for convex polygons, but here it's 720? Wait, maybe a typo, but if the graph is a horizontal line (constant (y) as (x) changes), then it's linear (slope 0). Wait, but the question is about how many graphs represent linear relationships. Wait, maybe the second graph is a horizontal line (so linear), and the first is not. Wait, but maybe I made a mistake. Wait, let's re - evaluate.

Wait, the first graph: points ((3,0)), ((4,2)), ((5,5)), ((8,20)). The differences in (y) over differences in (x) are not constant. The second graph: if it's a horizontal line (e.g., (y = 720) for all (x)), then the slope is 0 (constant), so it's linear. Wait, but maybe there's another graph? Wait, the original problem shows two graphs? Wait, the user's image: first graph (Number of Sides vs # of Diagonals), second graph (maybe related to angle sum). Wait, the sum of interior angles of a regular polygon is ((n - 2)\times180), which is a linear function of (n) (slope 180). Wait, maybe the second graph is the sum of interior angles. Let's check: for (n = 8), ((8 - 2)\times180=1080)? No, 720? Wait, ((6 - 2)\times180 = 720). Wait, maybe (n = 6). Wait, if the second graph is a linear graph (since ((n - 2)\times180) is linear in (n)), and the first is quadratic. Wait, but the first graph's points: ((3,0)): (\frac{3(3 - 3)}{2}=0), ((4,2)): (\frac{4(4 - 3)}{2}=2), ((5,5)): (\frac{5(5 - 3)}{2}=5), ((8,20)): (\frac{8(8 - 3)}{2}=20), which is quadratic. The second graph: if it's a linear function (like sum of interior angles, which is linear in (n)), then it's linear. Wait, but the question is "how many of the graphs represent a linear relationship". Wait, maybe one of them is linear. Wait, no, wait: the sum of exterior angles of any convex polygon is (360^\circ) (constant), so if the second graph is a horizontal line (exterior angles), it's linear (slope 0). But the sum of interior angles is linear. Wait, maybe the second graph is linear and the first is not. Wait, but the answer choices are 0,1,2,3. Wait, maybe I missed a graph. Wait, the user's image: maybe there are three graphs? Wait, the first graph (Number of Sides vs # of Diagonals), second graph (maybe related to angle sum), third? Wait, no, the user's image shows two graphs? Wait, the first graph has points at (3,0), (4,2), (5,5), (8,20). The second graph has a point at the end with y = 720. Wait, maybe the second graph is a horizontal line (so linear), and the first is not. But maybe the second graph is linear and the first is not, so only 1 graph is linear.

Answer:

1 (corresponding to the option "1")