Math Problem Statement
Let ABCD be a parallelogram. Points M and N are defined such that \( \overrightarrow{DM} = \frac{1}{3} \overrightarrow{DC} \) and \( \overrightarrow{BN} = 3 \overrightarrow{BC} \). The tasks are:
1. Draw an appropriate figure.
2. Show that \( \overrightarrow{AM} = \overrightarrow{BC} + \overrightarrow{DM} \).
3. Prove that \( \overrightarrow{AN} = 3 \overrightarrow{BC} + 3 \overrightarrow{DM} \).
4. Deduce that points A, M, and N are collinear.
Solution
Let's break down the given problem. The exercise revolves around a parallelogram and two points and defined by specific vector relations. Here's the breakdown:
Given Data:
Questions:
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Construire une figure convenable (Construct an appropriate figure):
You should draw the parallelogram and the points and following the given vector relations.- For , place it on the line segment such that . This means divides in a 1:2 ratio.
- For , we have , so lies on the line and extends beyond (since the vector is 3 times longer).
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**Montrer que
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Math Problem Analysis
Mathematical Concepts
Vector Geometry
Parallelogram Properties
Vector Addition
Formulas
Vector addition: \( \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \)
Collinearity condition for vectors
Theorems
Parallelogram vector properties
Suitable Grade Level
Grades 10-12