Congruent Triangles: Solving For Sides

Hey guys! Today, we're diving into a cool problem involving congruent triangles and a bit of algebra. We've got two triangles, let's call them triangle ABC and triangle DEF. The problem tells us that these triangles are congruent according to the Side-Side-Side (SSS) postulate, which is often referred to as LLL (Lado, Lado, Lado) in some regions. This means that all three sides of triangle ABC are equal in length to the corresponding three sides of triangle DEF. Knowing this, and given the side lengths in terms of $x$ and $y$, we're going to figure out the values of $x$ and $y$, and ultimately, the lengths of the sides.

Understanding Congruent Triangles

Before we jump into the math, let's make sure we're all on the same page about what it means for triangles to be congruent. In simple terms, congruent triangles are identical twins. They have the same size and the same shape. Imagine you cut out two triangles from a piece of paper, and they perfectly overlap when you put one on top of the other. That's congruence! For triangles, there are a few ways to prove congruence, like Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). Each of these postulates gives us a specific set of conditions that, if met, guarantee that the triangles are congruent. Our problem specifically mentions the SSS postulate, which is super handy because it only involves the side lengths.

When two triangles are congruent, it means that their corresponding parts (sides and angles) are equal. So, if triangle ABC is congruent to triangle DEF, then side AB is equal in length to side DE, side BC is equal to side EF, side CA is equal to side FD, angle A is equal to angle D, angle B is equal to angle E, and angle C is equal to angle F. This one-to-one correspondence is essential for solving problems involving congruent triangles. Remember, the order matters! If we say triangle ABC is congruent to triangle DEF, then A corresponds to D, B corresponds to E, and C corresponds to F.

In our case, we're given that the triangles are congruent by SSS, and we're given the lengths of some of the sides in terms of $x$ and $y$. This gives us a system of equations that we can solve to find the values of $x$ and $y$. Once we have those values, we can plug them back into the expressions for the side lengths to find the actual lengths of the sides. This is a classic example of how geometry and algebra can work together to solve problems. So, let's put on our thinking caps and get ready to solve some equations!

Setting up the Equations

The problem states that triangle ABC has a side BC with a length of 21 cm and a side AB with a length of $-2x + 3y$. Triangle DEF has a side ED with a length of $x + y$. Since the triangles are congruent by the LLL (SSS) postulate, we know that the corresponding sides must be equal in length. However, we aren't explicitly told which side of triangle DEF corresponds to side BC of triangle ABC. Let's consider the possibilities. Side AB corresponds to ED. Therefore: $-2x + 3y = x + y$. This is our first equation. We need another equation to solve for both $x$ and $y$. Let's assume that another side, EF, corresponds to side BC, and EF = $5x - 4y$. This gives us our second equation: $5x - 4y = 21$. So, we have the following system of equations:

• $-2x + 3y = x + y$
• $5x - 4y = 21$

Let's simplify the first equation: $-2x + 3y = x + y$. Subtracting $x$ from both sides gives $-3x + 3y = y$. Subtracting $3y$ from both sides gives $-3x = -2y$. Multiply both sides by -1: $3x = 2y$. We can express $y$ in terms of $x$: $y = \frac{3}{2}x$.

Now we substitute this expression for $y$ into the second equation: $5x - 4y = 21$. Replacing $y$ with $\frac{3}{2}x$, we get $5x - 4(\frac{3}{2}x) = 21$. Simplifying, we have $5x - 6x = 21$, which leads to $-x = 21$. Therefore, $x = -21$.

Next, we substitute the value of $x$ back into the expression for $y$: $y = \frac{3}{2}x = \frac{3}{2}(-21) = -\frac{63}{2} = -31.5$.

So we have $x = -21$ and $y = -31.5$. Now, let's check these values by plugging them back into our original equations.

Solving for x and y

Now that we have our system of equations, let's solve for $x$ and $y$. Here's our system again:

1. $-2x + 3y = x + y$
2. $5x - 4y = 21$

First, let's simplify equation (1). We can rewrite it as:

$-3x + 2y = 0$

Which gives us:

$2y = 3x$

$y = \frac{3}{2}x$

Now, substitute this value of $y$ into equation (2):

$5x - 4(\frac{3}{2}x) = 21$

$5x - 6x = 21$

$-x = 21$

So, $x = -21$. Now, plug this value back into the equation for $y$:

$y = \frac{3}{2}(-21)$

$y = -\frac{63}{2}$

$y = -31.5$

So, we have $x = -21$ and $y = -31.5$. These are the values of our variables! It's always a good idea to double-check our work, so let's plug these values back into the original equations to make sure they hold true.

Verifying the Solution

Let's verify our solution by plugging $x = -21$ and $y = -31.5$ back into the original equations:

Equation 1: $-2x + 3y = x + y$

$-2(-21) + 3(-31.5) = -21 + (-31.5)$

$42 - 94.5 = -52.5$

$-52.5 = -52.5$ (This checks out!)

Equation 2: $5x - 4y = 21$

$5(-21) - 4(-31.5) = 21$

$-105 + 126 = 21$

$21 = 21$ (This also checks out!)

Since both equations hold true, we can be confident that our values for $x$ and $y$ are correct. Great job, team! Now that we have the values of $x$ and $y$, we can find the lengths of the sides of the triangles.

Finding the Side Lengths

Now that we know $x = -21$ and $y = -31.5$, we can calculate the lengths of the sides of the triangles. We were given the following expressions for the side lengths:

• $AB = -2x + 3y$
• $ED = x + y$
• $BC = 21$

Let's plug in the values of $x$ and $y$ to find the lengths of sides AB and ED:

$AB = -2(-21) + 3(-31.5) = 42 - 94.5 = -52.5$

$ED = -21 + (-31.5) = -52.5$

Wait a minute! We have a problem. Side lengths cannot be negative. This indicates that there may be an error or inconsistency in the problem statement or our interpretation of it. It's possible that the expressions for the side lengths are not correct, or that the correspondence between the sides of the two triangles is different from what we assumed. Given that BC = 21, let's assume ED = 21 instead of solving the system of equations above. If $ED = x + y = 21$ and $AB = -2x + 3y$. We still need the equation, $5x - 4y = 21$ or any other side measure. Without having EF side we can not find the solution, because we will have 3 variables.

Conclusion

So, while we successfully solved for $x$ and $y$ using the given equations, the negative side lengths indicate a potential issue with the problem's setup. Always remember to check if your answers make sense in the context of the problem! It might be that we need more information, or that there's a mistake in how the problem was presented. Keep those critical thinking skills sharp, folks! In a real-world scenario, if you encounter a similar situation, it's crucial to re-evaluate the given information and assumptions to ensure the solution is valid. And that's a wrap for today's triangle adventure! Remember to always double-check your work and stay curious. See you in the next problem!