Proving A Geometric Relationship On A Line Segment A Detailed Solution

This article delves into a geometrical problem involving a line segment, its midpoint, and a point located on the line. We will explore the relationships between the distances of these points and prove a specific equation. This problem is a great exercise in understanding basic geometrical concepts and applying them to solve problems. The core of the problem revolves around the properties of line segments, midpoints, and distances between points on a line. By carefully analyzing the given information and applying the definitions, we can arrive at the desired conclusion. The problem also highlights the importance of visualizing the geometrical configuration to aid in problem-solving. We will use the given information about the lengths of the segments and the relative positions of the points to establish the relationship between the distances.

Let's consider the statement we need to analyze and solve.

a. Consider a segment $AB$ of length 5 cm, with point $O$ as its midpoint, and a point $M$ located on the line $AB$ such that $B$ is situated between $O$ and $M$, and $MB = 4$ cm. Prove that $2 \_ OM = MA + MB$.

This problem requires us to demonstrate a specific relationship between the distances of points on a line segment. The key elements of the problem are the line segment AB, its midpoint O, and another point M located on the same line. The position of M is crucial, as it is given that B lies between O and M. We are given the length of AB as 5 cm and the length of MB as 4 cm. The task is to prove that twice the length of OM is equal to the sum of the lengths of MA and MB. To solve this problem, we will utilize the properties of midpoints and line segments, along with the given lengths, to establish the required relationship. The problem tests our understanding of basic geometrical concepts and our ability to apply them in a proof.

To solve this problem, we will break it down into smaller steps, using the given information and geometrical principles to arrive at the desired conclusion. Let's begin by outlining the steps we will take:

1. Determine the lengths of $OA$ and $OB$ using the midpoint property.
2. Calculate the length of $OM$ using the given information about the position of point $B$ and the length of $MB$.
3. Calculate the length of $MA$ by considering the relationship between the segments $MB$, $AB$, and $MA$.
4. Finally, substitute the calculated lengths into the equation $2 \cdot OM = MA + MB$ and verify if it holds true.

Now, let's proceed with the calculations.

Step 1: Determine the lengths of $OA$ and $OB$

Since $O$ is the midpoint of the segment $AB$, it divides the segment into two equal parts. Given that the length of $AB$ is 5 cm, we can deduce that:

$OA = OB = \frac{AB}{2} = \frac{5}{2} = 2.5 \text{ cm}$

Therefore, the lengths of both $OA$ and $OB$ are 2.5 cm.

Step 2: Calculate the length of $OM$

We are given that $B$ is located between $O$ and $M$, and the length of $MB$ is 4 cm. We have already determined that the length of $OB$ is 2.5 cm. Since $OM$ is the sum of $OB$ and $MB$, we can calculate the length of $OM$ as follows:

$OM = OB + MB = 2.5 \text{ cm} + 4 \text{ cm} = 6.5 \text{ cm}$

Thus, the length of $OM$ is 6.5 cm.

Step 3: Calculate the length of $MA$

To find the length of $MA$, we need to consider the relationship between the segments. We know that $M$ lies on the line $AB$, and $B$ lies between $O$ and $M$. Therefore, we can express $MA$ as the sum of $MB$ and $BA$:

$MA = MB + BA$

We are given $MB = 4 \text{ cm}$ and $AB = 5 \text{ cm}$, so:

$MA = 4 \text{ cm} + 5 \text{ cm} = 9 \text{ cm}$

Therefore, the length of $MA$ is 9 cm.

Step 4: Verify the equation $2 \cdot OM = MA + MB$

Now, we have all the necessary lengths to verify the given equation. We have:

• $OM = 6.5 \text{ cm}$
• $MA = 9 \text{ cm}$
• $MB = 4 \text{ cm}$

Substituting these values into the equation, we get:

$2 \cdot OM = 2 \cdot 6.5 \text{ cm} = 13 \text{ cm}$

$MA + MB = 9 \text{ cm} + 4 \text{ cm} = 13 \text{ cm}$

Since $2 \cdot OM = 13 \text{ cm}$ and $MA + MB = 13 \text{ cm}$, the equation $2 \cdot OM = MA + MB$ holds true.

In conclusion, by carefully applying the definitions of midpoint and line segment properties, and using the given lengths, we have successfully demonstrated that $2 \cdot OM = MA + MB$. This problem highlights the importance of visualizing geometrical relationships and using them to solve problems. The solution involved breaking down the problem into smaller steps, calculating the necessary lengths, and then verifying the equation. This approach is a common strategy in problem-solving and can be applied to a wide range of geometrical problems. The result confirms the relationship between the distances of the points on the line segment, as stated in the problem.

The key to solving this problem was to understand the relationships between the different segments. By recognizing that O is the midpoint of AB, we could easily determine the lengths of OA and OB. Then, using the information about the position of M and the length of MB, we calculated OM. Finally, by considering the overall arrangement of the points, we found MA and verified the equation. This step-by-step approach, combined with a clear understanding of geometrical principles, allowed us to solve the problem effectively. The problem serves as a good example of how basic geometrical concepts can be used to solve more complex problems.

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