Complete the indirect proof.

Given: Rectangle JKLM has an area of 36 square centimeters. Side JK is at least 4 centimeters long.
Prove: KL<=9 centimeters

Statements:

Let
p: A=36 ∧ JK≥4
q: KL≤9

Indirect proof (by contradiction):
We try to establish:
p->q
i.e. if p is true, then q is true.

We will assume that p is true and q is false, i.e.
A=36, JK≥4, and that KL>9.

Let JK=4+x : x≥0, and
KL=9+y : y>0
so
A=JK*KL
=(4+x)(9+y)
=36+9x+4y+xy
Since y>0, so A>36
which cannot be true.
Therefore the assumption of
p and true and q is false is not valid (contradiction).
Therefore when p is true, q is true, or
p->q.
or
if A=36, and JK≥4 then KL≤9.

To prove that KL is less than or equal to 9 centimeters, we can use an indirect proof.

Assume, for the sake of contradiction, that KL is greater than 9 centimeters.

Since JKLM is a rectangle, opposite sides are equal in length. Therefore, JM is also greater than 9 centimeters.

Since the area of the rectangle JKLM is 36 square centimeters, we can use the formula for the area of a rectangle: area = length × width.

Let the length of JK be x centimeters. Since JK is at least 4 centimeters long, x is greater than or equal to 4.

Therefore, the width of the rectangle LK must be 36/x centimeters. Since KL is greater than 9 centimeters, LK must also be greater than 9 centimeters.

Using the area formula, JK × LK = 36, we can substitute the values:
x × (36/x) = 36
36 = 36

This equation is true for all positive values of x, including when x is equal to 4. Therefore, it does not contradict our assumption that KL is greater than 9 centimeters.

However, this contradicts our assumption that KL is greater than 9 centimeters. Therefore, our assumption was incorrect, and KL must be less than or equal to 9 centimeters.

Hence, we have successfully completed the indirect proof, and we have proven that KL is less than or equal to 9 centimeters.

To complete an indirect proof, we need to assume the opposite of what we want to prove and show that it leads to a contradiction or an impossible situation.

1. Assume for the sake of contradiction that KL is greater than 9 centimeters. (Assumption)
2. Since JKLM is a rectangle, KL is equal to JM, and the area of the rectangle is 36 square centimeters.
3. Let JM be x centimeters. Therefore, the area of the rectangle is x * JK = 36 square centimeters.
4. Since JK is at least 4 centimeters long, we have x * 4 = 36.
5. Solving for x, we find x = 9.
6. This means JM and KL are both equal to 9 centimeters, contradicting our assumption that KL is greater than 9 centimeters.
7. Therefore, our assumption is false, and KL must be less than or equal to 9 centimeters.

Hence, we have proven that KL is less than or equal to 9 centimeters using an indirect proof.