RS=8y+5, ST=5y+8, and RT=91.

Question

a. what is the value of y?
b. find RS and St

Answer 1

Given a triangle RST, by the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Using this, we can say that RS + ST > RT, so:

8y + 5 + 5y + 8 > 91
Combine like terms:
13y + 13 > 91
Subtract 13 from both sides:
13y > 78
Divide by 13:
y > 6

However, y must also satisfy ST + RT > RS, which translates to:

5y + 8 + 91 > 8y + 5
Combine like terms:
96 > 3y
Divide by 3:
y < 32

Therefore y is a number that is greater than 6 and less than 32.

From these inequalities, it cannot be determined a specific value for y.

b. The length of RS and ST depends on the exact value of y, which as we found out from the above calculation, cannot be determined specificly. However, we can express RS and ST in terms of y using the given expressions:

RS = 8y + 5
ST = 5y + 8.