Finding Y: Distance Between Two Points Is 13

by Alex Johnson 45 views

Let's dive into a fun math problem! We're going to find the values of y that make the distance between the points (1, y) and (-4, -5) exactly 13 units. Sounds like a plan? Let's get started!

Understanding the Distance Formula

Before we jump into solving for y, let's quickly refresh our memory about the distance formula. The distance formula is derived from the Pythagorean theorem and helps us calculate the distance between two points in a coordinate plane. If we have two points, say (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the distance d between them is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

This formula essentially calculates the length of the hypotenuse of a right triangle, where the legs are the differences in the x-coordinates and y-coordinates. It's a handy tool for many geometry problems, and it's exactly what we need for this problem.

Applying the Distance Formula to Our Problem

Now that we have the distance formula in mind, let's apply it to our specific problem. We are given the points (1, y) and (-4, -5), and we know that the distance between them is 13 units. Plugging these values into the distance formula, we get:

13=(−4−1)2+(−5−y)213 = \sqrt{(-4 - 1)^2 + (-5 - y)^2}

Our goal is to solve for y. To do this, we'll first square both sides of the equation to get rid of the square root:

132=(−4−1)2+(−5−y)213^2 = (-4 - 1)^2 + (-5 - y)^2

169=(−5)2+(−5−y)2169 = (-5)^2 + (-5 - y)^2

169=25+(−5−y)2169 = 25 + (-5 - y)^2

Next, subtract 25 from both sides:

144=(−5−y)2144 = (-5 - y)^2

Now, we can take the square root of both sides. Remember that when we take the square root, we need to consider both the positive and negative roots:

±12=−5−y\pm 12 = -5 - y

Solving for y

Now we have two separate equations to solve for y:

  1. 12=−5−y12 = -5 - y
  2. −12=−5−y-12 = -5 - y

Let's solve the first equation:

12=−5−y12 = -5 - y

Add 5 to both sides:

17=−y17 = -y

Multiply by -1:

y=−17y = -17

Now, let's solve the second equation:

−12=−5−y-12 = -5 - y

Add 5 to both sides:

−7=−y-7 = -y

Multiply by -1:

y=7y = 7

So, we have two possible values for y: -17 and 7.

Verifying Our Solutions

It's always a good idea to check our solutions to make sure they're correct. Let's plug each value of y back into the original distance formula to see if we get a distance of 13.

Checking y = -17

d=(−4−1)2+(−5−(−17))2d = \sqrt{(-4 - 1)^2 + (-5 - (-17))^2}

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

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

d=169d = \sqrt{169}

d=13d = 13

So, y = -17 is a valid solution.

Checking y = 7

d=(−4−1)2+(−5−7)2d = \sqrt{(-4 - 1)^2 + (-5 - 7)^2}

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

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

d=169d = \sqrt{169}

d=13d = 13

So, y = 7 is also a valid solution. Both solutions check out!

Conclusion

We have successfully found the values of y such that the distance between the points (1, y) and (-4, -5) is 13 units. The values are y = -17 and y = 7. Remember, the distance formula is a powerful tool derived from the Pythagorean theorem, and it helps us solve problems involving distances between points in the coordinate plane.

So, the values of yy are -17 and 7. These values satisfy the condition that the distance between the point (1,y)(1, y) and (−4,−5)(-4, -5) is 13 units.

And remember, mathematics is not just about formulas and calculations, it's about understanding the underlying concepts and using them to solve problems in a logical and systematic way.

For further reading and to deepen your understanding of the distance formula and its applications, consider visiting Khan Academy's Distance Formula resource.