Isosceles Trapezoid Area Calculation: A Step-by-Step Guide
Hey guys! Today, we're diving into a geometry problem that involves calculating the area of an isosceles trapezoid. We've got a classic scenario here, and we're going to break it down step by step so you can conquer similar problems with confidence. Let's jump right in!
Understanding the Problem
First off, let's restate the problem clearly. We have an isosceles trapezoid ABCD. In this trapezoid, the lengths of the equal sides (|AD| and |BC|) are 13cm. The height DE of the trapezoid, along with the length of the shorter base CD, is 12cm. Our mission, should we choose to accept it, is to calculate the area of this trapezoid ABCD. This problem combines geometric properties with area calculation, so it’s a great exercise in applying what we know about shapes and formulas.
Before we get into the nitty-gritty calculations, let’s visualize what we’re dealing with. An isosceles trapezoid is a four-sided figure (a quadrilateral) where two sides (the bases) are parallel, and the other two sides (the legs) are of equal length. This symmetry is key because it gives us some useful properties that we can exploit. In our case, AB and CD are the bases, with AB being the longer base, and AD and BC are the legs, both measuring 13cm. The height DE is perpendicular to both bases, forming right angles, which will be crucial for using the Pythagorean theorem later on.
Now, why is it important to understand each component of the trapezoid? Well, the formula for the area of a trapezoid involves the lengths of both bases and the height. So, to find the area, we need to figure out the length of the longer base AB. This is where the given information about the height and the equal sides comes into play. We're going to use these pieces of information to dissect the trapezoid into simpler shapes—rectangles and right triangles—that we can easily work with. Think of it like a puzzle: we have the pieces; we just need to arrange them correctly to reveal the solution.
Breaking Down the Trapezoid
Now, let's strategize how to tackle this problem. The key to solving this problem is to decompose the trapezoid into simpler geometric shapes. By drawing the height DE, we’ve essentially created a right-angled triangle (ADE) on one side and a rectangle (DECF, where F is the foot of the perpendicular from C to AB) in the middle. Since ABCD is an isosceles trapezoid, if we were to draw another height from C to AB (let's call the point where it meets AB as F), we'd create another congruent right-angled triangle (CFB) on the other side. This symmetry is super helpful because it means both triangles ADE and CFB are identical. Understanding this symmetry simplifies our calculations and gives us a clear path forward.
So, what does this decomposition give us? We now have a rectangle (DECF) whose length is the same as the shorter base CD (12cm) and whose height is the given height DE (12cm). We also have two identical right-angled triangles (ADE and CFB). To find the area of the trapezoid, we need to find the area of the rectangle and the areas of the two triangles, then add them all up. The area of the rectangle is straightforward: length times width. But the area of the triangles requires a bit more work. We need to find the length of the base AE (or FB, since they are the same). This is where the Pythagorean theorem comes to the rescue!
Why the Pythagorean theorem? Because we have a right-angled triangle (ADE), we know the length of the hypotenuse (AD = 13cm) and one of the legs (DE = 12cm). The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In simpler terms: a² + b² = c², where c is the hypotenuse, and a and b are the other two sides. We can use this to find the length of AE, which is the base of our triangle and a crucial piece of the puzzle.
Applying the Pythagorean Theorem
Alright, let's get our math hats on and apply the Pythagorean theorem to triangle ADE. We know that AD is the hypotenuse, with a length of 13cm, and DE is one of the legs, with a length of 12cm. We want to find the length of AE, which we'll call 'x'. So, using the Pythagorean theorem (a² + b² = c²), we can write the equation as:
x² + 12² = 13²
First, let's calculate the squares: 12² is 144, and 13² is 169. So our equation becomes:
x² + 144 = 169
Now, we need to isolate x². To do this, we subtract 144 from both sides of the equation:
x² = 169 - 144
x² = 25
Great! Now we have x² = 25. To find x, we need to take the square root of both sides:
x = √25
x = 5
So, the length of AE is 5cm. Remember, AE is one of the segments that makes up the longer base AB of the trapezoid. This is a significant step because it brings us closer to finding the total length of the longer base, which we need to calculate the area of the trapezoid.
Now that we know AE, we can leverage the symmetry of the isosceles trapezoid. Since the trapezoid is isosceles, the triangle on the other side (CFB) is congruent to triangle ADE. This means that FB also has a length of 5cm. This symmetry not only simplifies our calculations but also adds a layer of elegance to the problem-solving process. Understanding and utilizing symmetry is a powerful tool in geometry and can save us a lot of time and effort.
Calculating the Longer Base and Area
Now that we've found the length of AE (and FB) to be 5cm, we're in the home stretch! We need to calculate the length of the longer base AB. Remember that the trapezoid is made up of a rectangle (DECF) and two right-angled triangles (ADE and CFB). The longer base AB is composed of three segments: AE, EF, and FB. We know AE and FB are both 5cm. What about EF? Well, EF is the same length as the shorter base CD, which is given as 12cm. Therefore:
AB = AE + EF + FB
Substitute the values we know:
AB = 5cm + 12cm + 5cm
AB = 22cm
Fantastic! We've calculated the length of the longer base AB to be 22cm. Now we have all the pieces we need to calculate the area of the trapezoid. We know the lengths of both bases (AB = 22cm and CD = 12cm) and the height (DE = 12cm). The formula for the area of a trapezoid is:
Area = (1/2) * (base1 + base2) * height
In our case, base1 is AB (22cm), base2 is CD (12cm), and the height is DE (12cm). Plug these values into the formula:
Area = (1/2) * (22cm + 12cm) * 12cm
First, add the lengths of the bases:
Area = (1/2) * (34cm) * 12cm
Now, multiply by the height:
Area = (1/2) * 408cm²
Finally, multiply by 1/2:
Area = 204cm²
Final Answer
Boom! We've done it! The area of the isosceles trapezoid ABCD is 204 square centimeters. This problem was a fantastic journey through geometry, combining the properties of isosceles trapezoids, right-angled triangles, and the Pythagorean theorem. We broke down a complex shape into simpler components, applied relevant formulas, and methodically solved for the unknown. Remember, the key to tackling geometry problems is to visualize, decompose, and conquer!
So, there you have it, folks! We've successfully calculated the area of the isosceles trapezoid. Geometry problems like these might seem daunting at first, but with a clear strategy and a step-by-step approach, they become manageable and even enjoyable. Keep practicing, keep exploring, and remember that every problem solved is a step forward in mastering the world of math! Keep your questions coming, and let’s keep learning together!