Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 7740

45 sides

Step-by-step explanation: The sum of the measures of the interior angles of a polygon can be represented by the formula 180 (n - 2). Since the sum of the measures of the interior angles of the given polygon is 7,740°, we can set our formula equal to 7,740 so we have 180 (n - 2) = 7,740.

The n in the formula represents the number of sides of the polygon and the original question is asking for the number of sides of the polygon. So if we solve this equation for n, we will have our answer.

To solve the equation, we start by distributing the 180 through the parentheses on the left side and we have 180n - 360 = 7740 and adding 360 to both sides gives us 180n = 8,100. Dividing both sides of the equation by 180, we find that n = 45.

So if the sum of the measures of the interior angles of a polygon is 7740°, then we know that the polygon has 45 sides.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240