Finding The Side Length Of A Rhombus A Step-by-Step Solution

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Hey guys! Today, we're diving into a fun geometry problem that involves finding the side length of a rhombus. We're given some information about its area and diagonals, and we'll use that to crack the case. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's break down what we know. We have a rhombus, which is a quadrilateral with all four sides equal in length. We're given two key pieces of information:

  • The area of the rhombus is 336 cm².
  • The sum of the lengths of its diagonals is 62 cm.

Our mission, should we choose to accept it (and we do!), is to find the length of one side of this rhombus. To do this, we'll need to dust off some formulas and use a bit of algebraic manipulation. Don't worry, it's not as scary as it sounds!

Key Concepts for Solving the Rhombus Problem

Alright, let's arm ourselves with the knowledge we need to tackle this problem. Here are the key concepts we'll be using:

  1. Area of a Rhombus: The area of a rhombus can be calculated using the lengths of its diagonals. If we call the diagonals d1 and d2, the area (A) is given by:

    A = (1/2) * d1 * d2

    This formula is crucial because it connects the area we know (336 cm²) to the diagonals, which we have some information about (their sum is 62 cm). Understanding this relationship is the first step in unraveling the problem. The diagonals of a rhombus are perpendicular bisectors of each other, meaning they intersect at a right angle and divide each other into two equal parts. This property is essential for deriving the area formula and for understanding the geometry of the rhombus.

  2. Relationship Between Diagonals and Sides: The diagonals of a rhombus bisect each other at right angles, forming four congruent right-angled triangles. The sides of the rhombus are the hypotenuses of these triangles, and half the lengths of the diagonals are the legs. This is where the Pythagorean theorem comes into play! The fact that the diagonals intersect at right angles is super important. It allows us to use the Pythagorean theorem, which we'll see in a bit, to relate the side length to the diagonals.

  3. Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If we call the hypotenuse 'c' and the other sides 'a' and 'b', the theorem states:

c² = a² + b²

*This theorem will allow us to link the side length of the rhombus to the halves of the diagonals, which form the legs of the right triangles.* This connection is the key to finding the side length once we know the lengths of the diagonals. Remember, the sides of the rhombus are the hypotenuses of these right triangles, and half the diagonals are the legs.
  1. Algebraic Manipulation: We'll need to use our algebra skills to solve equations and find the values of the unknowns (the lengths of the diagonals and the side length). This might involve substitution, simplification, and solving quadratic equations. Don't worry, we'll take it step by step. Algebraic manipulation is the toolbox we'll use to actually solve for the unknowns. It's like having the right wrenches and screwdrivers to take apart and reassemble the problem.

Setting Up the Equations for Our Rhombus Problem

Okay, now that we have our concepts in place, let's translate the given information into mathematical equations. This is a crucial step in solving any word problem. Let's break it down:

  1. Define Variables:

    Let's use some variables to represent the unknowns:

    • Let d1 and d2 be the lengths of the diagonals of the rhombus.
    • Let 's' be the length of a side of the rhombus (this is what we want to find!).

    Defining variables is like labeling the parts of a machine before you start working on it. It makes everything much clearer. Using variables helps us keep track of what we're looking for and how the different pieces of information relate to each other. It's a foundational step in translating the word problem into a solvable mathematical form.

  2. Equation from the Area:

    We know the area of the rhombus is 336 cm², and we have the formula A = (1/2) * d1 * d2. So, we can write our first equation:

    (1/2) * d1 * d2 = 336

    Multiplying both sides by 2, we get:

    d1 * d2 = 672

    This equation connects the two diagonals through the given area. It's a vital piece of the puzzle. This equation is the direct translation of the area information into a mathematical statement. It tells us that the product of the diagonals is a specific number, which gives us a crucial relationship to work with.

  3. Equation from the Sum of Diagonals:

    We're told that the sum of the lengths of the diagonals is 62 cm. This gives us our second equation:

    d1 + d2 = 62

    This equation provides another connection between the two diagonals. Now we have two equations with two unknowns! This equation is the mathematical representation of the given sum of the diagonals. Together with the area equation, it forms a system of equations that we can solve to find the individual lengths of the diagonals.

  4. Equation from the Pythagorean Theorem:

    Remember those right-angled triangles formed by the diagonals? The sides of the rhombus are the hypotenuses, and half the diagonals are the legs. So, using the Pythagorean theorem:

    s² = (d1/2)² + (d2/2)²

    s² = (d1²/4) + (d2²/4)

    This equation links the side length 's' to the diagonals. It's the bridge that will allow us to find 's' once we know d1 and d2. This equation is the key to unlocking the side length. It connects the side we want to find to the diagonals, which we can solve for using our other equations. Remember, the diagonals bisect each other at right angles, creating those crucial right triangles.

Solving the Equations

Alright, we've set up our equations, and now it's time to put on our algebra hats and solve them! This is where things get interesting. We have three equations:

  1. d1 * d2 = 672
  2. d1 + d2 = 62
  3. s² = (d1²/4) + (d2²/4)

Our goal is to find 's', the side length. To do this, we first need to find d1 and d2.

  1. Solve for One Variable in Terms of the Other:

    Let's solve equation (2) for d2:

    d2 = 62 - d1

    This is a classic move in solving systems of equations. We isolate one variable so we can substitute it into another equation. Solving for one variable in terms of the other allows us to reduce the system of two equations with two unknowns into a single equation with one unknown, which is much easier to solve.

  2. Substitute into the Other Equation:

    Now, substitute this expression for d2 into equation (1):

    d1 * (62 - d1) = 672

    Expanding this, we get:

    62d1 - d1² = 672

    Rearranging to form a quadratic equation:

    d1² - 62d1 + 672 = 0

    We've transformed our system of equations into a single quadratic equation. Now we just need to solve it! This quadratic equation is the key to finding the lengths of the diagonals. It encapsulates the relationships between the diagonals and the given information, and solving it will give us the values of d1.

  3. Solve the Quadratic Equation:

    We can solve this quadratic equation by factoring, using the quadratic formula, or completing the square. Let's try factoring:

    (d1 - 48)(d1 - 14) = 0

    So, d1 = 48 or d1 = 14

    Excellent! We have two possible values for d1. This means we'll have two possible pairs of diagonals. Factoring the quadratic equation is an efficient way to find the solutions, which represent the possible lengths of one of the diagonals. The fact that we have two solutions indicates that there are two possible rhombuses that fit the given conditions.

  4. Find the Corresponding Values of d2:

    If d1 = 48, then d2 = 62 - 48 = 14

    If d1 = 14, then d2 = 62 - 14 = 48

    Notice that the values just switch! This makes sense because the equations are symmetric with respect to d1 and d2.

    We've found the two possible pairs of diagonal lengths. Now we can move on to finding the side length. We now have the lengths of both diagonals for the two possible rhombuses. This is a significant step forward, as we can now use this information to calculate the side length using the Pythagorean theorem.

  5. Calculate the Side Length 's':

    Now, we use equation (3) and either pair of d1 and d2 values (it won't matter which pair we use because of the symmetry):

    s² = (48²/4) + (14²/4)

    s² = (2304/4) + (196/4)

    s² = 576 + 49

    s² = 625

    s = √625

    s = 25

    There we have it! The side length of the rhombus is 25 cm. We've successfully used the diagonal lengths and the Pythagorean theorem to find the side length of the rhombus. This is the final step in solving the problem!

The Answer

So, the length of the side of the rhombus is 25 cm. We did it! We took a seemingly complex geometry problem and broke it down into manageable steps. We used our knowledge of rhombus properties, the Pythagorean theorem, and algebraic manipulation to find the solution.

Conclusion

Geometry problems can be a fun challenge, and this rhombus problem is a great example of how different mathematical concepts can come together to solve a single question. Remember, the key is to break down the problem, identify the relevant formulas and relationships, and then use your algebraic skills to find the unknowns. Keep practicing, and you'll become a geometry whiz in no time! And that's a wrap for today, guys! Hope you enjoyed this step-by-step journey through the rhombus problem. Until next time, keep those brains buzzing!