Paralelogramo: Perímetro, Base, Diagonal Y Área
Hey guys, let's dive into the fascinating world of parallelograms! We've got a cool problem on our hands today that involves finding the measure of the smaller diagonal and the area of a parallelogram. This isn't just about crunching numbers; it's about understanding the properties of these awesome shapes and how to apply them. So, buckle up, grab your calculators, and let's get this math party started!
Understanding the Basics of a Parallelogram
Alright, first things first, what exactly is a parallelogram? In simple terms, it's a flat, four-sided shape where opposite sides are parallel and equal in length. Think of a tilted rectangle, and you're pretty much there. This means that if you have a parallelogram ABCD, side AB is parallel to side DC, and side AD is parallel to side BC. Also, the length of AB is the same as DC, and the length of AD is the same as BC. Pretty straightforward, right? Now, this shape has some neat properties that we'll be using. For starters, opposite angles are equal, and consecutive angles add up to 180 degrees. So, if one angle is, say, 60 degrees, the opposite one is also 60 degrees, and the other two angles will be 120 degrees each (180 - 60 = 120). The perimeter of a parallelogram is just the total length around its edges, which is twice the sum of the lengths of adjacent sides. If we denote the base as 'b' and the oblique side as 's', the perimeter 'P' is given by P = 2(b + s). When we talk about the base and the oblique side, these are just adjacent sides of the parallelogram. The diagonal is a line segment connecting two non-adjacent vertices. A parallelogram has two diagonals, and they can have different lengths. Our mission today is to find the shorter one, and also to calculate the area of this particular parallelogram. The area can be calculated in a few ways, but a common formula is base times height (Area = b * h). We'll explore how to find that height! The problem gives us a specific scenario: a parallelogram with a perimeter of 48 cm. We're also told that the base measures 3 cm less than double the oblique side. This is a crucial piece of information that will help us determine the actual lengths of the sides. And, to top it all off, we know that one of the angles formed by the base and the oblique side is 60 degrees. This angle is super important because it allows us to use trigonometry to find heights and diagonals. So, we've got all the ingredients for a fantastic geometric puzzle!
Calculating the Lengths of the Sides
Okay, let's get down to business and figure out the actual lengths of our parallelogram's sides. We're given that the perimeter is 48 cm. Remember our formula for the perimeter? It's P = 2(b + s), where 'b' is the base and 's' is the oblique side. So, we have 48 = 2(b + s). If we divide both sides by 2, we get b + s = 24 cm. This tells us the sum of the base and the oblique side is 24 cm. Now, the problem also gives us a relationship between the base and the oblique side: "the base measures 3 cm less than the double of the oblique side". Let's translate this into an equation. If 's' is the oblique side, then double the oblique side is 2s. Three less than that is 2s - 3. So, our equation is b = 2s - 3. Brilliant! Now we have a system of two linear equations with two variables:
- b + s = 24
- b = 2s - 3
We can use substitution to solve this. Since equation 2 already gives us 'b' in terms of 's', let's substitute (2s - 3) for 'b' in equation 1:
(2s - 3) + s = 24
Now, let's simplify and solve for 's':
3s - 3 = 24
Add 3 to both sides:
3s = 27
Divide by 3:
s = 9 cm
Awesome! We've found the length of the oblique side. Now we can easily find the base using either equation. Let's use b = 2s - 3:
b = 2(9) - 3 b = 18 - 3 b = 15 cm
So, the base (b) is 15 cm and the oblique side (s) is 9 cm. Let's quickly check if these lengths give us the correct perimeter: 2(15 + 9) = 2(24) = 48 cm. Perfect! We've successfully cracked the side lengths. This is a huge step towards finding the diagonal and the area. Keep up the great work, guys!
Finding the Length of the Smaller Diagonal
Now that we know the lengths of the sides (base = 15 cm, oblique side = 9 cm) and one of the angles (60 degrees), we can finally find the measure of the smaller diagonal. Remember, a parallelogram has two diagonals. Let's visualize our parallelogram. Let the vertices be A, B, C, and D, starting from the bottom left and going counter-clockwise. Let AB be the base (15 cm) and AD be the oblique side (9 cm). The angle DAB is given as 60 degrees. The diagonals are AC and BD. We need to figure out which one is shorter. Generally, the diagonal opposite the smaller angle between the sides tends to be shorter. In our case, the angle between the 15 cm side and the 9 cm side is 60 degrees. The other angle between these sides (at vertex D, for example) would be 180 - 60 = 120 degrees. So, the diagonal connecting the vertices with the 120-degree angles is likely the shorter one. Let's assume we're looking for the diagonal BD. We can use the Law of Cosines to find the length of a diagonal in a triangle. If we consider triangle ABD, we have two sides (AB = 15 cm, AD = 9 cm) and the included angle (angle DAB = 60 degrees). Let's call the diagonal BD = d1.
The Law of Cosines states: c² = a² + b² - 2ab cos(C), where C is the angle opposite side c.
Applying this to our triangle ABD, with d1 as the side opposite the 60-degree angle:
d1² = AB² + AD² - 2(AB)(AD)cos(60°) d1² = 15² + 9² - 2(15)(9)cos(60°)
We know that cos(60°) = 0.5 (or 1/2).
d1² = 225 + 81 - 2(15)(9)(0.5) d1² = 306 - 270(0.5) d1² = 306 - 135 d1² = 171
To find d1, we take the square root of 171:
d1 = √171
d1 ≈ 13.08 cm
This is one of the diagonals. What about the other diagonal, AC? Let's call it d2. To find d2, we can consider triangle ABC. We know AB = 15 cm and BC = AD = 9 cm. The angle ABC is supplementary to angle DAB, so angle ABC = 180° - 60° = 120°. Applying the Law of Cosines to triangle ABC:
d2² = AB² + BC² - 2(AB)(BC)cos(120°) d2² = 15² + 9² - 2(15)(9)cos(120°)
We know that cos(120°) = -0.5 (or -1/2).
d2² = 225 + 81 - 2(15)(9)(-0.5) d2² = 306 - 270(-0.5) d2² = 306 + 135 d2² = 441
To find d2, we take the square root of 441:
d2 = √441 d2 = 21 cm
Comparing the two diagonals, d1 ≈ 13.08 cm and d2 = 21 cm. Clearly, d1 is the smaller diagonal. So, the measure of the smaller diagonal is approximately 13.08 cm (or √171 cm for the exact value). We're on a roll, guys! We've nailed the side lengths and one of the diagonals. Next up: the area!
Calculating the Area of the Parallelogram
Alright, team, we're in the home stretch! We've calculated the side lengths (base = 15 cm, oblique side = 9 cm) and the smaller diagonal (≈ 13.08 cm). Now, let's tackle the area of the parallelogram. There are a couple of ways we can do this, but a really straightforward method involves using the height. Remember the formula: Area = base × height. We know the base is 15 cm, but we need to find the height. The height is the perpendicular distance from the top side to the base. We can find this height using trigonometry, thanks to the 60-degree angle!
Let's go back to triangle ABD, where AB is the base (15 cm), AD is the oblique side (9 cm), and the angle DAB is 60 degrees. If we drop a perpendicular line from vertex D to the base AB (let's call the point where it meets the base E), we form a right-angled triangle ADE. In this right-angled triangle, AD is the hypotenuse (9 cm), and the angle DAE is 60 degrees. The height (h) of the parallelogram is the side DE.
In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse (sin(θ) = opposite/hypotenuse).
So, for angle DAE (60 degrees):
sin(60°) = DE / AD
We know that sin(60°) = √3 / 2 (or approximately 0.866).
Therefore:
√3 / 2 = h / 9
Now, we can solve for the height 'h':
h = 9 * (√3 / 2) h = (9√3) / 2 cm
This is the exact height. If you need an approximate value, √3 is about 1.732, so h ≈ 9 * (1.732 / 2) = 9 * 0.866 ≈ 7.794 cm.
Now that we have the height, we can calculate the area:
Area = base × height Area = 15 cm × (9√3 / 2) cm Area = (135√3) / 2 cm²
If you want the approximate area:
Area ≈ 15 cm × 7.794 cm Area ≈ 116.91 cm²
So, the area of the parallelogram is (135√3) / 2 cm², or approximately 116.91 cm². We did it, guys! We successfully calculated the lengths of the sides, the measure of the smaller diagonal, and the area of the parallelogram. It's amazing what you can do by understanding the properties of shapes and using a bit of trigonometry. Keep practicing, and you'll become math wizards in no time!
Conclusion: Mastering Parallelograms
Wow, what a journey! We started with a seemingly complex problem about a parallelogram and broke it down step by step. We figured out the lengths of the base and the oblique side using the given perimeter and the relationship between the sides. Then, we employed the Law of Cosines to determine the measure of the smaller diagonal, which required us to consider both possible diagonals and compare their lengths. Finally, we used trigonometry to find the height and subsequently calculate the area of the parallelogram. Each step built upon the last, showcasing how interconnected geometric concepts are. This problem highlights the importance of clearly defining variables, setting up equations correctly, and knowing which mathematical tools (like the perimeter formula, substitution, Law of Cosines, and trigonometric ratios) to apply. Remember, guys, math is all about problem-solving and logical thinking. The more you practice, the more confident you'll become in tackling different types of problems. So, whether you're dealing with parallelograms, triangles, or any other shape, the key is to stay curious, break down the problem, and have fun with the process. Keep exploring, keep learning, and you'll definitely conquer any math challenge that comes your way!