Tangent Lengths: Find X For Equal PA And PB
Hey guys! Let's dive into a cool geometry problem where we're dealing with circles and tangents. This is the kind of stuff that might seem tricky at first, but once you get the hang of it, it's super manageable. So, stick with me, and we'll break it down step by step. Our main goal here is to find the value of 'x' that makes two tangent segments equal in length. Ready? Let's get started!
Setting Up the Problem
First off, let's visualize what we're working with. Imagine a circle, and then picture a point 'P' hanging out somewhere outside that circle. From this point 'P', we're drawing two lines that just kiss the circle at points 'A' and 'B'. These lines, PA and PB, are what we call tangents. A crucial property of tangents drawn from the same external point to a circle is that they are equal in length. This is a fundamental concept in geometry, and it's exactly what we need to solve our problem.
Now, we're given that PA = 5x - 7 and PB = 2x + 20. The question asks us to find the value of 'x' that makes these two tangents equal. Since we know that tangents from the same point are equal, we can set up a simple equation:
5x - 7 = 2x + 20
This equation is the key to unlocking our answer. It tells us that whatever 'x' is, when we plug it into both expressions, we should get the same length for both tangents. Solving for 'x' will give us the value that satisfies this condition.
Why are tangent lengths equal? The equality of tangent lengths from an external point to a circle is not just some arbitrary rule; it's rooted in the fundamental properties of circles and triangles. To understand why this is true, consider the radii drawn from the center of the circle to the points of tangency, A and B. Let's call the center of the circle 'O'. Now, we have two triangles: triangle OAP and triangle OBP. Both OA and OB are radii of the circle, so they are equal in length. Also, the angles OAP and OBP are right angles because a radius is always perpendicular to the tangent at the point of tangency. Furthermore, the side OP is common to both triangles. By the Right-hand side, Hypotenuse, Side (RHS) congruence criterion, triangle OAP is congruent to triangle OBP. Therefore, PA = PB because they are corresponding sides of congruent triangles. Understanding this underlying principle not only helps in remembering the property but also enhances problem-solving skills in more complex geometric scenarios.
Solving for x
Alright, let's solve the equation we set up earlier. Here it is again:
5x - 7 = 2x + 20
To solve for 'x', we need to get all the 'x' terms on one side of the equation and all the constant terms on the other side. Let's start by subtracting 2x from both sides:
5x - 2x - 7 = 2x - 2x + 20
This simplifies to:
3x - 7 = 20
Next, we want to isolate the 'x' term, so we'll add 7 to both sides:
3x - 7 + 7 = 20 + 7
This gives us:
3x = 27
Finally, to find 'x', we divide both sides by 3:
3x / 3 = 27 / 3
So, we get:
x = 9
Checking the answer: It's always a good idea to check your answer by plugging the value of x back into the original expressions for PA and PB. This ensures that our solution is correct.
PA = 5x - 7 = 5(9) - 7 = 45 - 7 = 38 PB = 2x + 20 = 2(9) + 20 = 18 + 20 = 38
Since PA and PB are equal when x = 9, our solution is correct. Verifying the solution is a crucial step in problem-solving, as it confirms that the algebraic manipulations are accurate and that the value of x satisfies the given conditions. This practice reinforces understanding and builds confidence in the solution.
Analyzing the Options
Now, let's look at the answer choices provided:
A) 3 B) 5 C) 7 D) 10
Oops! It seems like our calculated value of x = 9 isn't listed among the options. Let's carefully re-examine our calculations to ensure we didn't make any mistakes. This is a common scenario in exams, where a simple arithmetic error can lead to an incorrect answer. Double-checking each step is essential to avoid these kinds of mistakes.
Recalculating the value of x: Let's go back to our equation and meticulously recalculate each step:
5x - 7 = 2x + 20
Subtract 2x from both sides:
3x - 7 = 20
Add 7 to both sides:
3x = 27
Divide by 3:
x = 9
It seems our calculations were correct all along. The value of x that makes the tangents equal is indeed 9. The absence of this value among the options suggests there might be an error in the provided choices. In such cases, it's important to trust your work, especially when you've double-checked each step. It's possible that the question has a typo or that the answer choices are incorrect.
What to do when the correct answer isn't listed: If you encounter a situation where your calculated answer doesn't match any of the provided options, here are a few steps to consider:
- Double-Check Your Work: Review each step of your solution to ensure there are no arithmetic or algebraic errors. Pay close attention to signs, operations, and substitutions.
- Re-Read the Question: Make sure you fully understand what the question is asking. Sometimes, misinterpreting the question can lead to an incorrect approach.
- Consider Alternative Methods: If possible, try solving the problem using a different method to see if you arrive at the same answer.
- Trust Your Solution: If you've thoroughly checked your work and are confident in your solution, it's possible that the question or answer choices contain an error. In exam situations, mark the question and move on, returning to it later if time permits.
Conclusion
So, even though 9 wasn't one of the choices, we know that's the value that makes PA and PB equal. Keep practicing these kinds of problems, and you'll become a geometry whiz in no time! Remember, the key is to understand the properties and apply them correctly. You got this!