Graphing LD: Slope, Distance, & Finding Point M

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Graphing LD: Slope, Distance, & Finding Point M

Hey math enthusiasts! Let's dive into some coordinate geometry fun. We're going to explore the line segment LD on graph paper, where we have the points L(-2, 1) and D(4, -4). We'll find the slope, calculate the distance, and even locate a special point M on the line. Buckle up, guys, because this is going to be a fun ride!

a. What is the Slope of LD?

Alright, let's start with the slope. The slope of a line is a measure of its steepness and direction. It tells us how much the y-coordinate changes for every unit change in the x-coordinate. It's often referred to as "rise over run." To calculate the slope of LD, we'll use the slope formula, which is: slope = (y₂ - y₁) / (x₂ - x₁).

First, let's identify our coordinates: L(-2, 1) and D(4, -4). We can label these as (x₁, y₁) and (x₂, y₂), respectively. So, x₁ = -2, y₁ = 1, x₂ = 4, and y₂ = -4. Now, plug these values into the slope formula: slope = (-4 - 1) / (4 - (-2)). Simplify the equation. The equation will be: slope = -5 / 6. Therefore, the slope of the line segment LD is -5/6. This means that for every 6 units we move to the right (run), we move 5 units down (rise).

Now, let's visualize this. Imagine you're standing at point L and want to get to point D. You'd need to go down and to the right. The slope of -5/6 tells us exactly how to do that, and this indicates that the line is going downwards from left to right. Understanding the slope helps us grasp the relationship between the x and y coordinates on the line segment. Knowing the slope is a crucial piece of the puzzle. It gives us a sense of the steepness of the line. The slope also tells us whether a line is increasing, decreasing, horizontal, or vertical. In this instance, because the slope is negative, we know that the line LD decreases as it moves from left to right. This is an important concept in understanding linear equations, which is a fundamental concept in mathematics. Remember, the slope is a constant value for a straight line. No matter which two points you choose on LD, the slope will always be -5/6. Understanding the slope is also very useful in other mathematical concepts like calculus.

b. What is LD (the Distance from L to D)?

Awesome, let's move on to finding the distance between points L and D. To calculate the distance between two points in a coordinate plane, we use the distance formula. This formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. The distance formula is: distance = √((x₂ - x₁)² + (y₂ - y₁)²)

We already know the coordinates of L(-2, 1) and D(4, -4). Let's plug them into the distance formula. We will have: distance = √((4 - (-2))² + (-4 - 1)²). Then, simplify the equation. distance = √((6)² + (-5)²). Calculate the values. distance = √(36 + 25). Calculate the final distance. distance = √61. So, the distance between L and D is √61 units, which is approximately 7.81 units. Remember that the distance is always a positive value, as it represents the length of the line segment between the two points. The distance formula helps us measure the actual length of a line segment. It's a handy tool for various geometric problems. It provides a numerical value that tells us the space between two points in the coordinate system. Knowing the distance is useful in further calculations. In geometry, knowing the lengths of line segments helps to calculate areas, perimeters, and volumes. The distance formula is also helpful in real-world scenarios, like in navigation or engineering, for measuring distances.

c. Determine the Coordinates of a Point M on Line LD That is Twice as Far from Point L as It Is from Point D.

Okay, things are getting interesting! We need to find a point M on the line LD that has a specific relationship with points L and D. The point M is twice as far from L as it is from D. This means the distance from L to M is twice the distance from M to D. We can use the section formula to find the coordinates of point M. The section formula is used to find the coordinates of a point that divides a line segment in a given ratio.

Let's say the ratio is m:n. In our case, since M is twice as far from L as it is from D, the ratio is 2:1 (because LM = 2 * MD). The section formula is: M(x, y) = ((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n)). We know the coordinates of L(-2, 1) and D(4, -4). So, x₁ = -2, y₁ = 1, x₂ = 4, y₂ = -4, m = 2, and n = 1. Now, let's plug these values into the section formula: M(x, y) = ((2 * 4 + 1 * (-2)) / (2 + 1), (2 * (-4) + 1 * 1) / (2 + 1)). Simplify the equation. M(x, y) = ((8 - 2) / 3, (-8 + 1) / 3). Calculate the values. M(x, y) = (6/3, -7/3). Therefore, the coordinates of point M are (2, -7/3), which is approximately (2, -2.33). So, the point M divides the line segment LD. We have successfully found the coordinates of point M, which satisfies the given condition. This demonstrates the application of the section formula and allows us to divide a line segment into specific ratios. This concept is useful in various geometric problems and other applications.

d. Describe How You Would Verify Your Answer

Great job, guys! Now, let's verify our findings and ensure we're on the right track. There are a few ways to confirm that our calculations are correct and that point M is located as we expect it to be.

  • Calculate the Distances: We can calculate the distance LM and MD using the distance formula. If our calculations are correct, the distance LM should be twice the distance MD. We have the coordinates of L(-2, 1), M(2, -7/3), and D(4, -4). Let's calculate the distances. First, find LM: LM = √((2 - (-2))² + (-7/3 - 1)²) = √(4² + (-10/3)²) = √(16 + 100/9) = √(244/9) ≈ 5.14 units. Then, find MD: MD = √((4 - 2)² + (-4 - (-7/3))²) = √(2² + (-5/3)²) = √(4 + 25/9) = √(61/9) ≈ 2.57 units. Now, check the ratio. Does LM ≈ 2 * MD? Yes, approximately, 5.14 ≈ 2 * 2.57. Because the values are approximately equal, this confirms our calculations and our answer is valid.
  • Check the Slope: Since M lies on line LD, the slope between L and M, and M and D, should be the same as the slope of LD (-5/6). Calculate the slope of LM: slope = (-7/3 - 1) / (2 - (-2)) = (-10/3) / 4 = -10/12 = -5/6. Calculate the slope of MD: slope = (-4 - (-7/3)) / (4 - 2) = (-5/3) / 2 = -5/6. Because the slopes are equal, this verifies our calculations and confirms that M lies on the line LD. Because the slopes are the same, this is another proof that the point M is correct. Also, if you were to graph the points, you could visually verify that point M is indeed on the line segment LD.
  • Use a Graphing Calculator or Software: You can input the coordinates of L, D, and M into a graphing calculator or software like GeoGebra. This allows you to visually confirm that M lies on the line segment LD and that the distances match your calculations. This is a very useful tool, because this is a visual confirmation of your calculations.

By performing these verification steps, we can confidently assert the accuracy of our solutions. The process of verification is crucial for the understanding of this question.

Alright, that's a wrap! We've successfully calculated the slope, distance, found the point M, and verified our answer. Keep practicing, and you'll become a coordinate geometry pro in no time! Keep up the good work! And now we are done.