Geometry Proof: Find The Missing Reason

Hey guys! Let's dive into this geometry proof and figure out what's missing. We're given some information about line segments and we need to find the reason that justifies a particular step in the proof. Geometry can be a bit like detective work, so let's put on our thinking caps and solve this puzzle!

The Problem

Here’s the proof we’re working with:

Given: $L M=12 ; L Z=3 x ; M Z=x$

Prove: $L Z=9$

Our mission, should we choose to accept it, is to identify the missing reason for Statement 5. What allows us to go from 4x = 12 to x = 3? Let's break it down.

Analyzing the Steps

Before we jump to the answer, let's make sure we understand each step of the proof. This will help us identify the missing reason more easily.

1. Given: We start with the information that $LM = 12$, $LZ = 3x$, and $MZ = x$. This is our foundation.
2. Segment Addition Postulate: This postulate states that if you have two smaller segments that make up a larger segment, the sum of the smaller segments equals the length of the larger segment. In this case, $LZ + MZ = LM$.
3. Substitution Property of Equality: We substitute the given values of $LZ$, $MZ$, and $LM$ into the equation from step 2. So, $3x + x = 12$.
4. Simplify: We combine like terms on the left side of the equation: $3x + x$ becomes $4x$. Therefore, $4x = 12$.
5. Missing Reason: This is what we need to figure out! We go from $4x = 12$ to $x = 3$.
6. Substitution Property of Equality: Now that we know the value of $x$, we substitute it back into the equation $LZ = 3x$. So, $LZ = 3(3)$.
7. Simplify: Finally, we simplify $3(3)$ to get $LZ = 9$, which is what we wanted to prove.

Identifying the Missing Reason

Okay, guys, let's focus on step 5. What mathematical operation takes us from $4x = 12$ to $x = 3$? We're essentially isolating $x$ by getting rid of the coefficient 4. To do that, we need to perform the inverse operation of multiplication, which is division.

We're dividing both sides of the equation by 4:

$(4x) / 4 = 12 / 4$

This simplifies to:

$x = 3$

So, the reason that justifies this step is the Division Property of Equality. This property states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced.

The Complete Proof

Now that we've identified the missing reason, let's present the complete proof:

Given: $L M=12 ; L Z=3 x ; M Z=x$

Prove: $L Z=9$

Why the Division Property of Equality Matters

The Division Property of Equality is a fundamental concept in algebra and geometry. It ensures that when we solve equations, we maintain balance and arrive at correct solutions. Without this property, we couldn't confidently manipulate equations to isolate variables and find their values.

In the context of geometric proofs, this property is crucial for establishing relationships between different segments and angles. It allows us to use algebraic techniques to prove geometric statements, linking the world of numbers and equations to the world of shapes and figures.

Alternative Properties to Consider (and Why They Don't Fit)

Sometimes, it's helpful to consider other properties that might seem relevant but ultimately don't fit the situation. This helps us solidify our understanding of why the correct answer is indeed correct.

• Subtraction Property of Equality: This property states that if you subtract the same value from both sides of an equation, the equation remains balanced. While subtraction is a valid operation, it's not what we're doing to get from $4x = 12$ to $x = 3$.
• Multiplication Property of Equality: This property states that if you multiply both sides of an equation by the same value, the equation remains balanced. Again, multiplication isn't the operation we're using in this step.
• Addition Property of Equality: Similar to subtraction, this property involves adding the same value to both sides of an equation, which isn't applicable here.

By process of elimination, we can further confirm that the Division Property of Equality is the correct reason for step 5.

Conclusion

So, there you have it! The missing reason for line 5 in the proof is the Division Property of Equality. By understanding the properties of equality and carefully analyzing each step of the proof, we were able to solve this geometric puzzle. Keep practicing, and you'll become a geometry pro in no time!

Remember, geometry is all about logical reasoning and applying the right principles. Keep sharpening those skills, and you'll be able to tackle any proof that comes your way!