2. exercise 18, 19 page 188. determine if the following statements are true. if a statement is true, give a proof from the definitions. if a statement is false, give a counterexample.

(a) if r and s are rational numbers, then (r+s)/2 is rational.
(b) for all real numbers a and b, if a < b then a < (a+b)/2 < b

a. true b. true

Step-by-step explanation:

(a) If r and s are rational numbers, then (r+s)/2 is rational.

true

rational numbers can be expresed as fractions

let be r=a/b and s=c/d being a,b,c,d  integer numbers

d.a=e is an integer number because it's the product of two integers

b.c=f is an integer number because it's the product of two integers

e+f=g  is an integer number because it's the sum of two integers

b.d=h is an integer number because it's the product of two integers

2.h=i is an integer number because it's the product of two integers

g/i=j is an integer number because it's the quotient of two integers

then

(b) For all real numbers a and b, if a < b then a < (a+b)/2 < b

true

lets analyze 2a < (a+b)

then 2a < (a+b) is true

lets analyze (a+b) < 2b

then (a+b) < 2b is true

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

answer: the resulting cross section is a rectangle. solution: a plane parallel to the base of a triangular prism will intersect a cross section that is the same shape as its bases. so the cross section is a triangle.

step-by-step explanation: