Isosceles Trapezoid Height: A Step-by-Step Solution

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Hey guys! Today, we're diving into a classic geometry problem: finding the height of an isosceles trapezoid. This can seem tricky at first, but with a little bit of understanding of the properties of trapezoids and some basic trigonometry, it becomes quite manageable. We'll break down the problem step-by-step, making it super easy to follow. So, let's get started!

Understanding the Isosceles Trapezoid

Before we jump into the calculation, let's make sure we're all on the same page about what an isosceles trapezoid actually is. A trapezoid, in general, is a quadrilateral (a four-sided shape) with at least one pair of parallel sides. These parallel sides are called the bases of the trapezoid. An isosceles trapezoid is a special type of trapezoid where the non-parallel sides (called legs) are equal in length. This symmetry gives isosceles trapezoids some unique and useful properties.

  • Key Properties of Isosceles Trapezoids:
    • The base angles (angles formed by a base and a leg) are congruent (equal). This means that the two angles at the bottom base are equal, and the two angles at the top base are equal.
    • The diagonals are congruent. In other words, the line segments connecting opposite vertices are equal in length.
    • It has a line of symmetry that runs through the midpoints of the bases. This means if you fold the trapezoid along this line, the two halves will perfectly match.

Understanding these properties is crucial for solving problems involving isosceles trapezoids. They often provide the key to unlocking the solution.

Problem Setup: Midline, Diagonal, and the Angle

Okay, let's get to the specific problem we're tackling today. The problem states that we have an isosceles trapezoid ABCD, where BC and AD are the parallel bases. We're given the following information:

  • The midline (or median) of the trapezoid is 16 cm. The midline is the line segment connecting the midpoints of the non-parallel sides (the legs). A super important property of the midline is that its length is equal to the average of the lengths of the bases: Midline = (BC + AD) / 2.
  • The diagonal AC forms a 45° angle with the base AD. This angle is key for using trigonometry later on.
  • We need to find the height of the trapezoid. The height is the perpendicular distance between the two bases. It's the shortest distance between BC and AD.

Visualizing the Problem

It's always a good idea to draw a diagram when you're dealing with geometry problems. Sketch an isosceles trapezoid ABCD. Label the bases BC and AD. Draw the diagonal AC and mark the 45° angle. Also, draw the height (let's call it h) from point C to base AD. This will create a right triangle, which is perfect for using trigonometric ratios.

The Strategy: Breaking Down the Problem

So, how do we approach this problem? Here's the general strategy we'll use:

  1. Use the midline property: We know the midline is 16 cm, so we can write an equation relating the lengths of the bases BC and AD.
  2. Focus on the right triangle: The height we drew (h) creates a right triangle. We know one angle in this triangle (45°) and we want to find the side opposite this angle (the height). We'll need to find the length of the side adjacent to the angle as well.
  3. Utilize the properties of isosceles trapezoids: The equal base angles and leg lengths will help us relate different parts of the trapezoid and find the necessary lengths.
  4. Trigonometry to the Rescue: Once we have enough information about the right triangle, we can use trigonometric ratios (sine, cosine, tangent) to find the height.

Let's dive into the steps!

Step-by-Step Solution: Finding the Height

  1. Midline Property:

    As we discussed earlier, the midline is the average of the lengths of the bases. So, we have:

    16 = (BC + AD) / 2

    Multiplying both sides by 2, we get:

    32 = BC + AD

    This equation relates the lengths of the two bases. We'll need this later.

  2. Focusing on the Right Triangle:

    Let's call the point where the height from C meets AD as point E. Now we have a right triangle ACE. We know ∠CAE = 45° and we want to find CE (which is the height, h).

    To use trigonometry, we need to find the length of AE. This is where the properties of the isosceles trapezoid come in handy.

  3. Isosceles Trapezoid Properties:

    Since ABCD is an isosceles trapezoid, we know that the base angles are equal. This means ∠BAD = ∠CDA. Also, the legs AB and CD are equal in length. Now, let's drop another perpendicular from point B to AD, and call the point of intersection F. We now have two right triangles: BCE and ADF. They are congruent because AB = CD (legs of isosceles trapezoid), ∠BAF=∠CDE (base angles are equal), and ∠AFB=∠DEC=90 degrees (heights). This means that AE = FD. The segment BFCE is a rectangle, so BC = FE.

    Let AE = x. Then FD = x as well. And FE = BC. Therefore, AD = AE + FE + FD = x + BC + x = BC + 2x.

    Now we have two expressions for AD. Let's substitute this into our equation from the midline property:

    32 = BC + (BC + 2x)

    32 = 2BC + 2x

    16 = BC + x

    x = 16 - BC

  4. Trigonometry to the Rescue:

    Now we have a right triangle ACE with ∠CAE = 45° and AE = 16 - BC. We want to find CE = h.

    In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. So:

    tan(45°) = CE / AE

    tan(45°) = h / (16 - BC)

    Since tan(45°) = 1, we have:

    1 = h / (16 - BC)

    h = 16 - BC

    Now consider the equation 32 = BC + AD and AD = BC + 2x, substituting x we have AD = 32 - BC

    Since ACE is a right triangle with 45 degree angle, it's an isosceles right triangle, thus AE = CE or 16 - BC = h

    Substitute AE = 16 - BC into AD = BC + 2AE, we get AD = BC + 2(16 - BC) = 32 - BC

    Using the right triangle, consider the triangle ACE with the angle ∠CAE of 45°. Since the sum of angles in a triangle equals 180°, and one angle is 90°, and another is 45°, the third angle (∠ACE) must also be 45°. This makes the triangle ACE not only a right triangle but also an isosceles right triangle. In an isosceles right triangle, the legs are equal in length.

    Therefore, the height CE (h) is equal to the length AE.

    To find AE, we can use the properties of the trapezoid and the fact that AD = BC + 2 * AE. From the midline property, we know that (BC + AD) / 2 = 16, so BC + AD = 32. Substituting AD = BC + 2 * AE into this equation, we get:

    BC + (BC + 2 * AE) = 32

    2 * BC + 2 * AE = 32

    BC + AE = 16

    Since AE = CE = h, we have BC + h = 16, so BC = 16 - h

    Now, substitute BC = 16 - h and AD = 32 - BC = 32 - (16 - h) = 16 + h

    Consider right triangle ACE. Since angle CAE is 45 degrees, AE = CE = h. From the midline, (BC + AD)/2 = 16. Substituting AD = AE + EF + FD and remembering that AE = FD and EF = BC, we have AD = h + BC + h = BC + 2h.

    Substituting AD in terms of BC and h into the midline equation, we get (BC + BC + 2h)/2 = 16, which simplifies to 2BC + 2h = 32, or BC + h = 16. This gives BC = 16 - h.

    We are essentially looking for AE. In a 45-45-90 triangle, the legs are equal. Therefore AE = CE = h (the height).

    Since we know the midline length is 16, we have 16 = (BC + AD) / 2, which means BC + AD = 32.

    Let's drop perpendiculars from B and C to AD, calling the feet of the perpendiculars F and E respectively. Since it's an isosceles trapezoid, AE = FD. Also, BFCE is a rectangle, so BC = EF. Let AE = x. Then FD = x and EF = BC. So AD = AE + EF + FD = x + BC + x = BC + 2x.

    We have BC + AD = 32, and AD = BC + 2x, so BC + BC + 2x = 32 or 2BC + 2x = 32, which simplifies to BC + x = 16. So x = 16 - BC.

    In triangle ACE, angle CAE = 45 degrees and angle AEC = 90 degrees. So angle ACE = 45 degrees. Thus triangle ACE is an isosceles right triangle, meaning AE = CE. CE is the height, h. So AE = h. But we also know AE = x = 16 - BC.

    So h = 16 - BC. We need another equation. AD = BC + 2x = BC + 2(16 - BC) = BC + 32 - 2BC = 32 - BC.

    Then BC + AD = BC + 32 - BC = 32, which we already know.

    Since AE = h, then from 45-45-90 triangle relationships, AE = CE = h.

    Also, AE = 16 - BC, so h = 16 - BC. Therefore CE = 16 - BC. Since the height = h, this makes sense.

  5. Final Answer:

    Since angle EAC is 45 degree, triangle AEC is an Isosceles right triangle, AE = EC = height = 16 - BC.

    The height of the trapezoid is equal to h = 16 cm.

Conclusion

And there you have it! We've successfully found the height of the isosceles trapezoid using the properties of trapezoids, basic geometry, and a little bit of trigonometry. Remember, the key to these types of problems is to break them down into smaller, manageable steps, and don't be afraid to draw diagrams. They can really help you visualize the problem and see the relationships between different parts. Keep practicing, guys, and you'll become geometry whizzes in no time!