Finding Triangle Sides: A Math Problem Explained

by Admin 49 views
Finding Triangle Sides: A Math Problem Explained

Hey guys! Let's dive into a fun geometry problem. We're going to break down how to find the possible integer lengths of a side of a triangle. This is the kind of problem that can seem a bit tricky at first, but with a little bit of knowledge and some careful steps, we can totally crack it. We'll look at the conditions given, apply the triangle inequality theorem, and then figure out the whole number solutions that fit the criteria. Ready? Let's get started!

The Problem: Setting the Stage

Alright, so here's the deal. We're trying to build a triangle called ABC. We already know some important information: the length of side AB is 0.139 meters, and the length of side AC is 65 millimeters. Also, a key clue: if we divide the length of side BC by 5, there's a remainder of 3. What we want to figure out is: what are all the possible whole number lengths for side BC? And, of course, we need to justify our answer. No worries, we'll go through it step by step. We have to consider the basics of triangle formation and apply some cool math rules to solve this problem effectively. Understanding this will give you a solid foundation for more complex geometry problems in the future.

First off, we need to make sure all our measurements are in the same units. We have AB in meters and AC in millimeters. Let's convert everything to millimeters, since it's usually easier to work with whole numbers. Since 1 meter is equal to 1000 millimeters, AB is 0.139 * 1000 = 139 mm. Now we have: AB = 139 mm, AC = 65 mm. We will denote the length of BC as x.

So, the problem is about finding possible integer values for BC (represented as x), given that when x is divided by 5, the remainder is 3. This is our fundamental constraint that will guide us. Additionally, we need to consider the constraints imposed by triangle inequality, which is a key concept in triangle geometry. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This ensures that a triangle can actually be formed with the given side lengths. Failing to follow this leads to a situation where the sides can not be connected to form a triangle. So, applying this concept, the sum of lengths of AB and AC must be greater than BC, the sum of AB and BC must be greater than AC, and the sum of AC and BC must be greater than AB.

Therefore, we need to apply this theorem to the lengths of AB, AC, and BC, ensuring that the relationships satisfy the conditions required for a valid triangle to be constructed. Therefore, we should write these conditions mathematically to show these constraints correctly. These constraints will limit the range of possible values for side BC, which further makes the process straightforward. We will go through the proper steps to solve this problem, so you guys get it. The goal is to get the possible integer solutions for BC in the triangle. Let's dive in!

Applying the Triangle Inequality Theorem

Okay, so the triangle inequality theorem is our best friend here. It tells us that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is crucial; otherwise, you can't actually build a triangle – the sides won't be able to connect! This theorem is a fundamental concept in geometry, essential for understanding the properties and feasibility of triangles. It sets the rules that determine whether a set of side lengths can actually form a closed, three-sided figure. We'll apply this theorem to our triangle ABC.

So, let's break it down for our triangle:

  1. AB + AC > BC
  2. AB + BC > AC
  3. AC + BC > AB

We know that AB = 139 mm and AC = 65 mm. Let's substitute those values in and use x to represent the length of BC.

  1. 139 + 65 > x
  2. 139 + x > 65
  3. 65 + x > 139

Let's simplify these inequalities:

  1. 204 > x (This means x must be less than 204)
  2. x > -74 (This is always true since side lengths can't be negative)
  3. x > 74 (This means x must be greater than 74)

So, putting it all together, we have that x must be less than 204 and greater than 74. In other words, 74 < x < 204. This tells us the range of possible values for the length of side BC. But we're not done yet! We also have that x, when divided by 5, leaves a remainder of 3. That will further narrow down our options.

Therefore, by applying the theorem, we have set the boundaries of possible integer values for the side BC. This step is pivotal for understanding the possible lengths, which are constrained within the mathematical boundaries, allowing us to find the possible side values for side BC. We have now restricted the range of possible solutions to the problem, making the subsequent steps more straightforward for finding the precise integer values that fulfil the conditions of both triangle inequality and the remainder condition.

Finding the Possible Integer Values of BC

Alright, we now know that the length of BC (x) has to be between 74 and 204 (74 < x < 204), and when you divide it by 5, you get a remainder of 3. How do we find the numbers that fit those rules? Easy-peasy! We can use the remainder condition to narrow things down. The fact that the remainder is 3 tells us something specific about the number. That means that x can be written in the form of 5n + 3, where n is a whole number. So, x will be a multiple of 5 plus 3. We can write some numbers based on this pattern:

78 (515 + 3), 83 (516 + 3), 88 (517 + 3), 93 (518 + 3), 98 (519 + 3), 103 (520 + 3), 108 (521 + 3), 113 (522 + 3), 118 (523 + 3), 123 (524 + 3), 128 (525 + 3), 133 (526 + 3), 138 (527 + 3), 143 (528 + 3), 148 (529 + 3), 153 (530 + 3), 158 (531 + 3), 163 (532 + 3), 168 (533 + 3), 173 (534 + 3), 178 (535 + 3), 183 (536 + 3), 188 (537 + 3), 193 (538 + 3), 198 (5*39 + 3).

Now we need to check which of these numbers fall within the range we found from the triangle inequality (74 < x < 204). All of the numbers we generated meet this condition. So, the possible whole number lengths for side BC are: 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188, 193, and 198. These represent all the lengths that satisfy both the triangle inequality and the remainder constraint. These are the possible lengths of the side BC. This means we have successfully solved the problem using all the mathematical concepts!

By following this method, we can determine the possible values for the third side of the triangle, given the constraints of the triangle inequality and the remainder condition. The ability to combine multiple mathematical concepts provides a comprehensive answer to the problem, which enables us to determine the complete range of valid side lengths.

Conclusion: The Final Answer

So, in summary, to find the possible integer lengths of side BC, we first converted all measurements to the same units. Then we applied the triangle inequality theorem to find the range of possible lengths. Finally, we used the remainder condition to find specific integer values within that range that met the conditions. The possible integer lengths for side BC are: 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188, 193, and 198.

Hopefully, you guys found that helpful! Remember, when you're facing a geometry problem, always break it down step by step and make sure you understand the key concepts. And don't be afraid to draw diagrams to help visualize the problem. Keep practicing, and you'll get the hang of it! Feel free to ask if you have more questions.