Problem Analysis:
The image depicts a triangular arrangement of circles, with some numbers already filled in and others left blank. The goal is to determine the missing numbers such that the sum of the numbers along each side of the triangle equals the "Magic Number," which is given as
19.
Known Information:
1. The Magic Number is
19.
2. The numbers already filled in are:
- Bottom-left circle:
3
- Middle-left circle:
8
- Bottom-right circle:
4
Objective:
Determine the values of the remaining circles (top, middle-right, and bottom-middle) such that the sum of the numbers along each side of the triangle equals
19.
Triangle Sides:
The triangle has three sides:
1.
Left side: Top circle + Middle-left circle + Bottom-left circle.
2.
Right side: Top circle + Middle-right circle + Bottom-right circle.
3.
Bottom side: Bottom-left circle + Bottom-middle circle + Bottom-right circle.
Step-by-Step Solution:
#### 1. Define Variables:
Let:
- \( T \) be the value of the top circle.
- \( M_R \) be the value of the middle-right circle.
- \( B_M \) be the value of the bottom-middle circle.
#### 2. Write Equations for Each Side:
Using the Magic Number (
19) for each side:
-
Left side: \( T + 8 + 3 = 19 \)
-
Right side: \( T + M_R + 4 = 19 \)
-
Bottom side: \( 3 + B_M + 4 = 19 \)
#### 3. Solve Each Equation:
##### Left Side:
\[ T + 8 + 3 = 19 \]
\[ T + 11 = 19 \]
\[ T = 19 - 11 \]
\[ T = 8 \]
So, the top circle (\( T \)) is
8.
##### Bottom Side:
\[ 3 + B_M + 4 = 19 \]
\[ B_M + 7 = 19 \]
\[ B_M = 19 - 7 \]
\[ B_M = 12 \]
So, the bottom-middle circle (\( B_M \)) is
12.
##### Right Side:
\[ T + M_R + 4 = 19 \]
Substitute \( T = 8 \):
\[ 8 + M_R + 4 = 19 \]
\[ M_R + 12 = 19 \]
\[ M_R = 19 - 12 \]
\[ M_R = 7 \]
So, the middle-right circle (\( M_R \)) is
7.
#### 4. Verify the Solution:
-
Left side: \( T + 8 + 3 = 8 + 8 + 3 = 19 \) (Correct)
-
Right side: \( T + M_R + 4 = 8 + 7 + 4 = 19 \) (Correct)
-
Bottom side: \( 3 + B_M + 4 = 3 + 12 + 4 = 19 \) (Correct)
All sides satisfy the Magic Number condition.
Final Answer:
The missing numbers are:
- Top circle:
8
- Middle-right circle:
7
- Bottom-middle circle:
12
\[
\boxed{8, 7, 12}
\]
Parent Tip: Review the logic above to help your child master the concept of magic triangles worksheet.
