The screen in the auditorium is 3 times as long as the screen in the library and a width 3 times as much the screen in the library is 3 meters long with a perimeter of 10

The perimeter of the auditorium is 30 meters

Step-by-step explanation:

The question is

What is the perimeter of the screen in the auditorium?

we know that

If two figures are similar, then the ratio of its perimeters is equal to the scale factor

In this problem, the auditorium and the library are similar

Let

z ----> the scale factor

x ----> the perimeter of the auditorium

y ----> the perimeter of the library

so

we have

----> the scale factor is given

substitute

solve for x

therefore

The perimeter of the auditorium is 30 meters

For educational purposes

Find the width of the library

we know that

so

Find the dimensions of the auditorium

To find out the dimensions of the auditorium multiply by 3 (the scale factor) the dimensions of the library

so

Verify the perimeter

----> is correct

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).