B is the midpoint of line SL, Y is the midpoint of line SB and BI = 4. What does SB =? What is the coordinate of Y? Is B the midpoint of line YI? What is line YB congruent to? S is -6 and L is 18. Explain your answers

Let's break down the problem step by step.

### 1. **Find the length of line segment \( SL \)**

You are given that \( S = -6 \) and \( L = 18 \). The length of the line segment \( SL \) is the distance between these two points:

\[
SL = L - S = 18 - (-6) = 18 + 6 = 24
\]

### 2. **Find the length of line segment \( SB \)**

Since \( B \) is the midpoint of \( SL \), the length of \( SB \) is half the length of \( SL \):

\[
SB = \frac{SL}{2} = \frac{24}{2} = 12
\]

Thus, \( SB = 12 \).

### 3. **Find the coordinate of \( B \)**

The coordinate of the midpoint \( B \) of the line segment \( SL \) is the average of the coordinates of \( S \) and \( L \):

\[
B = \frac{S + L}{2} = \frac{-6 + 18}{2} = \frac{12}{2} = 6
\]

So, the coordinate of \( B \) is \( 6 \).

### 4. **Find the coordinate of \( Y \)**

\( Y \) is the midpoint of line \( SB \). Since the coordinates of \( S \) and \( B \) are \( -6 \) and \( 6 \), respectively, the coordinate of \( Y \) is:

\[
Y = \frac{S + B}{2} = \frac{-6 + 6}{2} = \frac{0}{2} = 0
\]

So, the coordinate of \( Y \) is \( 0 \).

### 5. **Is \( B \) the midpoint of line \( YI \)?**

You are given that \( BI = 4 \), so \( I \) is located 4 units away from \( B \). Since \( B = 6 \), the coordinate of \( I \) is either:

\[
I = B + 4 = 6 + 4 = 10 \quad \text{or} \quad I = B - 4 = 6 - 4 = 2
\]

Next, let's check if \( B \) is the midpoint of \( YI \). The midpoint of \( YI \) would be:

\[
\text{Midpoint of } YI = \frac{Y + I}{2}
\]

For \( I = 10 \):

\[
\frac{Y + I}{2} = \frac{0 + 10}{2} = 5
\]

For \( I = 2 \):

\[
\frac{Y + I}{2} = \frac{0 + 2}{2} = 1
\]

Since neither of these results give \( B = 6 \), \( B \) is **not** the midpoint of line \( YI \).

### 6. **What is line \( YB \) congruent to?**

Since \( Y \) is the midpoint of line segment \( SB \), the length of \( YB \) must be half the length of \( SB \):

\[
YB = \frac{SB}{2} = \frac{12}{2} = 6
\]

Thus, line segment \( YB \) is congruent to line segment \( YS \), as both are \( 6 \) units in length.

---

### Final Answers:
1. \( SB = 12 \)
2. The coordinate of \( Y \) is \( 0 \)
3. \( B \) is **not** the midpoint of line \( YI \)
4. Line \( YB \) is congruent to line \( YS \)

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. How do you calculate the midpoint of a line segment in 3D space?
2. How would the problem change if \( BI \) was 8 instead of 4?
3. How do you determine if two line segments are congruent?
4. What are the properties of midpoints in geometric constructions?
5. How do the coordinates change if \( S \) and \( L \) are 3D points instead of 1D?

**Tip:** To calculate the distance between two points, subtract their coordinates and take the absolute value!

B is the midpoint of line SL, Y is the midpoint of line SB and BI = 4. What does SB =? What is the coordinate of Y? Is B the midpoint of line YI? What is line YB congruent to? S is -6 and L is 18. Explain your answers.

Explore how to find the midpoints of line segments SL and SB, determine if B is the midpoint of YI, and find which segments are congruent. This problem involves applying the midpoint formula and checking segment congruence, suitable for Grades 7-10.

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