Solutions For Exercise 11 Distance Gradient And Midpoint Explained

Hey guys! Let's break down Exercise 11, where we're tackling distance, gradient, and midpoint calculations. This is super fundamental stuff in coordinate geometry, so let's get it right. We've got five pairs of points to work with, and for each pair, we need to find:

1. The distance between them.
2. The gradient (or slope) of the line joining them.
3. The midpoint of the line segment connecting them.

Ready? Let's dive in!

a. Points P(4, 2) and Q(12, 8)

i. Distance Between P and Q

Distance is the first concept we will tackle. To find the distance between two points, we use the distance formula, which is derived from the Pythagorean theorem. If you think about it, the difference in x-coordinates and the difference in y-coordinates form the two legs of a right triangle, and the distance between the points is the hypotenuse. The formula looks like this:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Here, P is $(x_1, y_1)$ and Q is $(x_2, y_2)$. Plugging in our values:

$$d = \sqrt{(12 - 4)^2 + (8 - 2)^2}$$

$$d = \sqrt{(8)^2 + (6)^2}$$

$$d = \sqrt{64 + 36}$$

$$d = \sqrt{100}$$

$$d = 10$$

So, the distance between points P and Q is 10 units. It's crucial to remember this formula; it's your best friend in coordinate geometry! The distance formula is a direct application of the Pythagorean theorem, linking algebra and geometry seamlessly. Understanding the geometric interpretation helps in memorizing and applying the formula correctly. Think of it as finding the length of the hypotenuse of a right triangle formed by the two points. The horizontal and vertical distances between the points are the legs of the triangle, and the distance we calculate is the hypotenuse. This visual connection not only aids memory but also provides a deeper understanding of the concept.

ii. Gradient of the Line Joining P and Q

Next up is the gradient, or slope, which tells us how steep the line is. The gradient formula is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Using our points P and Q:

$$m = \frac{8 - 2}{12 - 4}$$

$$m = \frac{6}{8}$$

$$m = \frac{3}{4}$$

The gradient of the line joining P and Q is $\frac{3}{4}$. This means that for every 4 units we move horizontally, we move 3 units vertically. The gradient is a measure of the steepness and direction of a line. A positive gradient indicates that the line slopes upwards from left to right, while a negative gradient means it slopes downwards. A gradient of zero represents a horizontal line, and an undefined gradient (division by zero) represents a vertical line. The magnitude of the gradient tells us how steep the line is; a larger magnitude indicates a steeper line. Understanding the concept of gradient is essential for various applications, including determining the rate of change in real-world scenarios. For example, in physics, the gradient can represent the velocity of an object if we plot distance against time.

iii. Midpoint of the Line Joining P and Q

Now, let's find the midpoint, which is the point exactly halfway between P and Q. The midpoint formula is:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Plugging in our coordinates:

$$M = \left( \frac{4 + 12}{2}, \frac{2 + 8}{2} \right)$$

$$M = \left( \frac{16}{2}, \frac{10}{2} \right)$$

$$M = (8, 5)$$

The midpoint of the line joining P and Q is (8, 5). Think of the midpoint as the average of the x-coordinates and the average of the y-coordinates. The midpoint formula is a straightforward way to find the exact center of a line segment. It is simply the average of the x-coordinates and the y-coordinates of the two endpoints. This concept is incredibly useful in geometry and various practical applications. For example, in computer graphics, the midpoint can be used to divide a line segment into two equal parts, which is crucial for creating smooth curves and shapes. In surveying and mapping, finding the midpoint helps in locating the center of a property or a region. Understanding the midpoint also lays the foundation for more advanced concepts, such as finding the centroid of a triangle or other geometric figures.

b. Points X(3, 3) and Y(7, 6)

i. Distance Between X and Y

Let's use the distance formula again:

$$d = \sqrt{(7 - 3)^2 + (6 - 3)^2}$$

$$d = \sqrt{(4)^2 + (3)^2}$$

$$d = \sqrt{16 + 9}$$

$$d = \sqrt{25}$$

$$d = 5$$

The distance between points X and Y is 5 units. Remember, the distance is always a positive value! The distance formula is a fundamental tool in coordinate geometry, allowing us to quantify the separation between two points in a Cartesian plane. Its derivation from the Pythagorean theorem highlights the deep connection between algebra and geometry. When applying the distance formula, it’s essential to ensure that you correctly identify the x and y coordinates of the two points. A common mistake is to subtract coordinates in the wrong order, but since we square the differences, the sign error gets nullified. However, consistent application of the formula with correct substitution is crucial for accurate results. Understanding the conceptual basis of the distance formula reinforces its importance and applicability in more complex geometric problems.

ii. Gradient of the Line Joining X and Y

Now, for the gradient:

$$m = \frac{6 - 3}{7 - 3}$$

$$m = \frac{3}{4}$$

The gradient of the line joining X and Y is $\frac{3}{4}$. This gradient is the same as the one we found for points P and Q, indicating that these lines are parallel. The gradient, often referred to as the slope, is a crucial concept in understanding the properties of lines. It quantifies the steepness and direction of a line, providing a measure of how much the y-coordinate changes for a unit change in the x-coordinate. A positive gradient indicates an upward slope from left to right, while a negative gradient signifies a downward slope. The magnitude of the gradient reflects the steepness of the line; a larger magnitude implies a steeper line. Parallel lines have equal gradients, and perpendicular lines have gradients that are negative reciprocals of each other. This understanding of gradients is fundamental in various applications, including calculus, physics, and engineering.

iii. Midpoint of the Line Joining X and Y

Using the midpoint formula:

$$M = \left( \frac{3 + 7}{2}, \frac{3 + 6}{2} \right)$$

$$M = \left( \frac{10}{2}, \frac{9}{2} \right)$$

$$M = (5, 4.5)$$

The midpoint of the line joining X and Y is (5, 4.5). Don't be afraid of decimals; midpoints can certainly have non-integer coordinates! The midpoint formula provides a straightforward method to find the point that lies exactly halfway between two given points. It is based on the simple concept of averaging the x-coordinates and the y-coordinates of the two points. The midpoint is a critical concept in various geometric constructions and applications. For instance, it is used to bisect a line segment, which is a fundamental operation in Euclidean geometry. In computer graphics and animation, the midpoint is often used to divide lines and curves into segments, facilitating smoother rendering and animations. Understanding the midpoint formula is also essential for advanced topics like coordinate geometry and analytical geometry.

c. Points R(3, 2) and S(9, 6)

i. Distance Between R and S

Let's calculate the distance:

$$d = \sqrt{(9 - 3)^2 + (6 - 2)^2}$$

$$d = \sqrt{(6)^2 + (4)^2}$$

$$d = \sqrt{36 + 16}$$

$$d = \sqrt{52}$$

$$d = 2\sqrt{13}$$

The distance between points R and S is $2\sqrt{13}$ units. Sometimes, you'll get answers with square roots; just simplify them as much as possible! The distance formula is a crucial tool in coordinate geometry, allowing us to quantify the separation between two points in a Cartesian plane. It is derived from the Pythagorean theorem, highlighting the relationship between algebra and geometry. When applying the distance formula, it’s essential to correctly identify and substitute the x and y coordinates of the two points. Common errors include incorrect subtraction order, but the squaring operation mitigates sign errors. However, consistent and accurate substitution is vital for precise results. The distance formula’s conceptual basis reinforces its significance and applicability in more complex geometric problems.

ii. Gradient of the Line Joining R and S

Let's find the gradient:

$$m = \frac{6 - 2}{9 - 3}$$

$$m = \frac{4}{6}$$

$$m = \frac{2}{3}$$

The gradient of the line joining R and S is $\frac{2}{3}$. This gradient tells us the line is less steep than the ones we saw earlier. The gradient, also known as the slope, is a fundamental concept in understanding the properties of lines. It quantifies the steepness and direction of a line, measuring the change in the y-coordinate for a unit change in the x-coordinate. A positive gradient indicates an upward slope from left to right, while a negative gradient signifies a downward slope. The magnitude of the gradient reflects the line's steepness; a larger magnitude implies a steeper line. Parallel lines share equal gradients, and perpendicular lines have gradients that are negative reciprocals of each other. This understanding of gradients is crucial in various applications, including calculus, physics, and engineering.

iii. Midpoint of the Line Joining R and S

Using the midpoint formula:

$$M = \left( \frac{3 + 9}{2}, \frac{2 + 6}{2} \right)$$

$$M = \left( \frac{12}{2}, \frac{8}{2} \right)$$

$$M = (6, 4)$$

The midpoint of the line joining R and S is (6, 4). The midpoint is always a single, unique point between the two endpoints. The midpoint formula offers a straightforward method to determine the point exactly halfway between two given points. It is based on averaging the x-coordinates and the y-coordinates of the two points. The midpoint concept is crucial in numerous geometric constructions and applications. For instance, it is used to bisect a line segment, a fundamental operation in Euclidean geometry. In computer graphics and animation, the midpoint is often utilized to divide lines and curves into segments, facilitating smoother rendering and animations. Understanding the midpoint formula is also essential for advanced topics in coordinate and analytical geometry.

d. Points A(4, -2) and B(-4, 2)

i. Distance Between A and B

Using the distance formula:

$$d = \sqrt{(-4 - 4)^2 + (2 - (-2))^2}$$

$$d = \sqrt{(-8)^2 + (4)^2}$$

$$d = \sqrt{64 + 16}$$

$$d = \sqrt{80}$$

$$d = 4\sqrt{5}$$

The distance between points A and B is $4\sqrt{5}$ units. Watch out for those negative signs; they can be tricky! The distance formula is a foundational tool in coordinate geometry, enabling us to measure the separation between two points in a Cartesian plane. Its derivation from the Pythagorean theorem underscores the connection between algebra and geometry. When applying the distance formula, precise identification and substitution of the x and y coordinates of the points are crucial. Common mistakes involve errors in subtraction, but the squaring operation mitigates sign issues. Nonetheless, consistent and accurate application ensures precise results. Understanding the distance formula’s conceptual basis reinforces its significance and application in more complex geometric scenarios.

ii. Gradient of the Line Joining A and B

Let's find the gradient:

$$m = \frac{2 - (-2)}{-4 - 4}$$

$$m = \frac{4}{-8}$$

$$m = -\frac{1}{2}$$

The gradient of the line joining A and B is $-\frac{1}{2}$. A negative gradient means the line slopes downwards from left to right. The gradient, or slope, is a fundamental concept for understanding the properties of lines. It quantifies the steepness and direction of a line, measuring the change in the y-coordinate for a unit change in the x-coordinate. A positive gradient indicates an upward slope from left to right, while a negative gradient indicates a downward slope. The magnitude of the gradient reflects the line's steepness; a larger magnitude implies a steeper line. Parallel lines share equal gradients, and perpendicular lines have gradients that are negative reciprocals of each other. This understanding of gradients is vital in various applications, including calculus, physics, and engineering.

iii. Midpoint of the Line Joining A and B

Using the midpoint formula:

$$M = \left( \frac{4 + (-4)}{2}, \frac{-2 + 2}{2} \right)$$

$$M = \left( \frac{0}{2}, \frac{0}{2} \right)$$

$$M = (0, 0)$$

The midpoint of the line joining A and B is (0, 0). This means the line segment AB is centered at the origin! The midpoint formula provides a simple method for determining the point exactly halfway between two given points. It involves averaging the x-coordinates and the y-coordinates of the two points. The midpoint concept is essential in various geometric constructions and applications. For example, it is used to bisect a line segment, which is a fundamental operation in Euclidean geometry. In computer graphics and animation, the midpoint is often utilized to divide lines and curves into segments, facilitating smoother rendering and animations. A solid grasp of the midpoint formula is also crucial for more advanced topics in coordinate and analytical geometry.

e. Points M(4, 8) and N(-1, -4)

i. Distance Between M and N

Let's calculate the distance:

$$d = \sqrt{(-1 - 4)^2 + (-4 - 8)^2}$$

$$d = \sqrt{(-5)^2 + (-12)^2}$$

$$d = \sqrt{25 + 144}$$

$$d = \sqrt{169}$$

$$d = 13$$

The distance between points M and N is 13 units. This one gives us a nice whole number! The distance formula is an indispensable tool in coordinate geometry, allowing us to quantify the separation between two points in a Cartesian plane. It is derived from the Pythagorean theorem, highlighting the interplay between algebra and geometry. When applying the distance formula, careful identification and substitution of the x and y coordinates of the points are essential. Common mistakes include sign errors during subtraction, but the squaring operation typically mitigates these. Consistent and accurate application of the formula is crucial for achieving precise results. Understanding the conceptual basis of the distance formula reinforces its significance and broad applicability in geometric problem-solving.

ii. Gradient of the Line Joining M and N

Now, for the gradient:

$$m = \frac{-4 - 8}{-1 - 4}$$

$$m = \frac{-12}{-5}$$

$$m = \frac{12}{5}$$

The gradient of the line joining M and N is $\frac{12}{5}$. This is a pretty steep line, guys! The gradient, also known as the slope, is a fundamental concept in understanding the properties of lines. It quantifies the steepness and direction of a line, measuring the change in the y-coordinate for a unit change in the x-coordinate. A positive gradient indicates an upward slope from left to right, while a negative gradient signifies a downward slope. The magnitude of the gradient reflects the line's steepness; a larger magnitude implies a steeper line. Parallel lines have equal gradients, and perpendicular lines have gradients that are negative reciprocals of each other. This understanding of gradients is vital in numerous applications, including calculus, physics, and engineering.

iii. Midpoint of the Line Joining M and N

Using the midpoint formula:

$$M = \left( \frac{4 + (-1)}{2}, \frac{8 + (-4)}{2} \right)$$

$$M = \left( \frac{3}{2}, \frac{4}{2} \right)$$

$$M = (1.5, 2)$$

The midpoint of the line joining M and N is (1.5, 2). And that's the last midpoint for this exercise! The midpoint formula provides a straightforward method for determining the point exactly halfway between two given points. It involves averaging the x-coordinates and the y-coordinates of the two points. The midpoint concept is essential in various geometric constructions and applications. For instance, it is used to bisect a line segment, which is a fundamental operation in Euclidean geometry. In computer graphics and animation, the midpoint is often utilized to divide lines and curves into segments, facilitating smoother rendering and animations. A solid grasp of the midpoint formula is also crucial for more advanced topics in coordinate and analytical geometry.

Conclusion

Alright, we've worked through all five pairs of points, finding the distance, gradient, and midpoint for each. You've nailed it if you followed along! Remember, these are fundamental concepts in coordinate geometry, and mastering them will set you up for success in more advanced topics. Keep practicing, and you'll become a pro in no time! These concepts—distance, gradient, and midpoint—form the bedrock of coordinate geometry and analytical geometry. The distance formula, derived from the Pythagorean theorem, allows us to measure the separation between two points in a Cartesian plane. The gradient, also known as the slope, quantifies the steepness and direction of a line, providing insights into its inclination. The midpoint formula offers a simple yet powerful method to find the exact center of a line segment. These tools are not only crucial in mathematics but also find extensive applications in physics, engineering, computer graphics, and various other fields. By mastering these basic concepts, you build a strong foundation for tackling more complex problems and real-world applications. Consistent practice and a deep understanding of the underlying principles will solidify your proficiency in coordinate geometry.