Isosceles Trapezoid Problem: Angles And Lengths Calculation

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Isosceles Trapezoid Problem: Angles and Lengths Calculation

Hey guys! Let's dive into a fun geometry problem involving an isosceles trapezoid. We've got an interesting setup here, and we're going to break it down step by step. So, grab your thinking caps, and let’s get started!

Problem Statement

We have an isosceles trapezoid ABCD where AB is parallel to CD. We know that AB = 6 cm, BC = 2√2 cm, angle C = 45°, and BE is perpendicular to DC, with E being a point on DC. Our mission, should we choose to accept it, is twofold:

  • a) Calculate the measures of the other angles of trapezoid ABCD.
  • b) Calculate the lengths of the segments EC, DE, and DC.

Sounds like a plan? Awesome! Let's break down each part.

Part a: Calculating the Other Angles

Alright, so the first part of our quest is to figure out the measures of all the angles in this trapezoid. Remember, an isosceles trapezoid has some cool properties that make our lives easier. Specifically, the base angles are equal. That means angle C is equal to angle D, and angle A is equal to angle B. Let's leverage that!

Understanding the Properties of an Isosceles Trapezoid

Before we jump into calculations, let's quickly recap the key properties of an isosceles trapezoid. This will help us build a solid foundation for solving the problem. Here are the essentials:

  1. Two parallel sides: These are the bases (AB and CD in our case).
  2. Two non-parallel sides (legs) are equal: BC = AD.
  3. Base angles are equal: ∠C = ∠D and ∠A = ∠B.
  4. The sum of angles on the same side (adjacent angles) is 180°: ∠A + ∠D = 180° and ∠B + ∠C = 180°.

With these properties in mind, we can start cracking the angle puzzle. Because the trapezoid ABCD is isosceles, we know that angle C is equal to angle D. We're given that angle C is 45 degrees, so angle D is also 45 degrees. See, that wasn't so hard, was it? Now we're halfway there!

To find angles A and B, we need to remember another key property: in a trapezoid, angles on the same side (adjacent angles) add up to 180 degrees. So, angle B + angle C = 180 degrees. We know angle C is 45 degrees, so we can set up a simple equation:

Angle B + 45° = 180°

Solving for angle B, we subtract 45 degrees from both sides:

Angle B = 180° - 45° = 135°

Since the trapezoid is isosceles, angle A is equal to angle B. Therefore, angle A is also 135 degrees. Boom! We've found all the angles.

Summarizing the Angles

Let's make it crystal clear what we've discovered:

  • Angle A = 135°
  • Angle B = 135°
  • Angle C = 45°
  • Angle D = 45°

So, the other angles of trapezoid ABCD are 135°, 135°, and 45°. We nailed it! Now, let's move on to the second part of the problem, where we'll calculate the lengths of some segments.

Part b: Calculating the Lengths of Segments EC, DE, and DC

Okay, time for the next challenge! We need to figure out the lengths of segments EC, DE, and DC. This part might seem a bit trickier, but don't worry, we'll tackle it step by step. Remember that perpendicular line BE? That’s going to be our best friend here.

Using the Right Triangle BEC

Since BE is perpendicular to DC, we've created a right triangle BEC. We know that angle C is 45 degrees, and we know the length of BC (2√2 cm). This is perfect! We can use trigonometry or the properties of a 45-45-90 triangle to find the length of EC.

In a 45-45-90 triangle, the two legs are equal in length, and the hypotenuse is √2 times the length of a leg. In our case, BC is the hypotenuse, and EC and BE are the legs. Let's denote the length of EC (and BE) as x. So, we have:

BC = x√2

We know BC = 2√2 cm, so:

2√2 = x√2

Divide both sides by √2:

x = 2 cm

Therefore, the length of EC is 2 cm. We've got one segment down! High five! This also means that the length of BE is 2 cm, which we might need later.

Finding DE

To find DE, we need to think a bit differently. Notice that since ABCD is an isosceles trapezoid, if we drop another perpendicular from A to DC (let's call the point of intersection F), we'll create another right triangle, AFD. Moreover, ABEF will be a rectangle. Why is this helpful? Well, in a rectangle, opposite sides are equal.

So, AB = FE = 6 cm. Also, triangles BEC and AFD are congruent (they are both 45-45-90 triangles with equal hypotenuses). This means that EC = FD = 2 cm.

Now we know FD, but we need DE. Think about it – in an isosceles trapezoid, the lengths DE and EC would be the same. However, to calculate DE, we can consider the triangle BEC and AFD. Since triangles AFD and BEC are congruent (by Angle-Angle-Side congruence), DE must be equal to EC. Thus, DE = 2 cm.

Calculating DC

Now for the grand finale: finding the length of DC. We know that DC is made up of three segments: DE, EC, and FE. We've already found the lengths of DE and EC (both 2 cm), and we know FE is the same as AB (6 cm). So, we can simply add them up:

DC = DE + EC + FE DC = 2 cm + 2 cm + 6 cm DC = 10 cm

And there you have it! The length of DC is 10 cm. We've successfully found the lengths of all the segments we needed!

Summarizing the Segment Lengths

Let's recap what we've calculated:

  • EC = 2 cm
  • DE = 2 cm
  • DC = 10 cm

Woohoo! We've conquered this trapezoid problem. Give yourselves a pat on the back – you've earned it!

Final Thoughts

So, guys, we've successfully navigated through this isosceles trapezoid problem, figuring out both the angles and the lengths of various segments. Geometry can be tricky, but with a clear understanding of the properties and a step-by-step approach, anything is possible! Keep practicing, and you'll become geometry wizards in no time. Until next time, keep those brains buzzing!