Finding The Radius: Isosceles Trapezoid & Inscribed Circle

Oct 22, 2025 by SLV Team 59 views

Finding the Radius of an Inscribed Circle in an Isosceles Trapezoid

Hey guys! Let's dive into a geometry problem involving an isosceles trapezoid and a circle inscribed within it. We'll be using some cool math concepts to figure out the radius of that circle. Specifically, we're going to tackle a problem where we have an isosceles trapezoid with bases AB = 8 and CD = 18. Our mission? To find the length of the radius of the inscribed circle. This is a classic geometry problem, so get ready to flex those math muscles! We will break down the steps, explaining everything clearly so you can follow along. No need to be a math whiz – we'll go slow and steady.

Understanding the Problem and Key Concepts

Okay, so what exactly are we dealing with? First, we have an isosceles trapezoid. This means we have a four-sided shape (a quadrilateral) where one pair of opposite sides (the bases) are parallel, and the other two sides (the legs) are equal in length. Think of a table with slanted sides – that's your isosceles trapezoid! In our case, the bases are labeled AB and CD, with lengths of 8 and 18, respectively. The bases are the parallel sides. Then, we have an inscribed circle. This is a circle that sits inside the trapezoid, touching all four sides. The cool thing about a circle inscribed in a trapezoid (or any polygon) is that it is tangent to each side. That's a fancy way of saying the circle touches each side at exactly one point, forming a 90-degree angle at the point of contact. The radius of this circle is the distance from the center of the circle to any of the sides. And that's what we want to find out! Now, the key idea here is that for a circle to be inscribed in a trapezoid, the sum of the lengths of the two parallel sides (the bases) must equal the sum of the lengths of the non-parallel sides (the legs). This is because the circle will be tangent to all the sides. So, the circumference of a circle inscribed in a trapezoid has a special relationship with the trapezoid's sides. Another helpful concept to keep in mind is the formula for the area of a trapezoid: Area = (1/2) * height * (base1 + base2). This will come in handy later. But let's not get ahead of ourselves. First, we need to understand the relationship between the bases and the legs when a circle is inscribed.

When we have an inscribed circle, the height of the trapezoid is equal to the diameter of the circle. This is because the height is the perpendicular distance between the bases, and the circle touches both bases. Furthermore, the legs of the isosceles trapezoid have a special relationship with the bases. This is because the circle is tangent to all four sides, creating symmetry and geometric relationships. Finally, the radius will be half of the diameter.

The Geometry of Isosceles Trapezoids and Inscribed Circles

Let's unpack this a bit further. The fact that the circle is tangent to all sides means that from the center of the circle, we can draw radii (plural of radius) that are perpendicular to each side. This creates right angles at the points where the circle touches the sides. Consider the height of the trapezoid. It’s the perpendicular distance between the bases. Since the circle touches both bases, the height of the trapezoid is actually the diameter of the circle. This is a super important point! Now, think about the legs of the trapezoid. Because the trapezoid is isosceles, the legs are equal in length. And since the circle is inscribed, these legs have a specific relationship with the bases, which helps us calculate the length of those legs. Specifically, the sum of the bases (AB + CD) equals the sum of the lengths of the legs. Knowing the lengths of the bases, we can determine the length of each leg. This will prove useful. The legs of the trapezoid create two right triangles with the height and a portion of the bases. These right triangles are congruent, due to the symmetry of the trapezoid. So, we can apply the Pythagorean theorem. With the height (which is the diameter) and part of the base, we can calculate the length of the leg.

Step-by-Step Solution

Alright, let's roll up our sleeves and get to the good stuff: solving the problem! We will break this problem down into manageable chunks. The goal is to make it easy for you to follow along.

1. Finding the Height

As mentioned earlier, the height of the trapezoid is equal to the diameter of the inscribed circle. But how do we find the height? Well, we can use the properties of the isosceles trapezoid and the inscribed circle. First, remember that in an isosceles trapezoid, the legs are equal in length. Also, the sum of the bases (AB + CD) equals the sum of the legs. This is because a circle inscribed in a trapezoid is tangent to all sides. So, AB + CD = 8 + 18 = 26. This also means that the sum of the lengths of the legs is 26. Since the legs are equal, each leg has a length of 26 / 2 = 13. Now, let’s consider the right triangles formed by dropping perpendiculars from the shorter base (AB) to the longer base (CD). These perpendiculars are the height of the trapezoid (let's call it 'h'). The base of each right triangle is (CD - AB) / 2 = (18 - 8) / 2 = 5. We now have a right triangle with a leg of length 5 and a hypotenuse (the leg of the trapezoid) of length 13. We can use the Pythagorean theorem to find the height: a² + b² = c². So, h² + 5² = 13². This gives us h² = 169 - 25 = 144. Therefore, h = √144 = 12. Since the height is the diameter, the diameter of the circle is 12.

2. Calculating the Radius

This is the easy part! The radius (r) of a circle is half its diameter (d). We found that the diameter is 12. Therefore, the radius is r = d / 2 = 12 / 2 = 6. Boom! We have our answer.

Conclusion: The Answer is 6!

So, the length of the radius of the circle inscribed in the isosceles trapezoid is 6. Congrats, you've successfully solved the problem! You used your knowledge of isosceles trapezoids, inscribed circles, and some basic geometry to find the solution. You did great! This problem is a great example of how different geometric concepts come together. Keep practicing, and you'll become a geometry whiz in no time. You have now seen how important it is to remember that the height of the trapezoid is the same as the diameter of the inscribed circle. Also, the legs of the trapezoid can be used to form right triangles, allowing us to use the Pythagorean theorem.

Key Takeaways and Tips

Recognize the properties: Always remember the unique characteristics of isosceles trapezoids and inscribed circles. Knowing that the height is the diameter and the relationship between sides is crucial. The sum of the bases always equals the sum of the legs when a circle is inscribed. Use this fact!
Draw Diagrams: Drawing a clear diagram can help you visualize the problem and identify the relationships between the different parts of the shape and the circle.
Break it down: Break down complex problems into smaller, more manageable steps. This will make the problem easier to solve.
Practice: Practice similar problems to reinforce your understanding and build confidence. The more you practice, the easier it will become.
Use the Pythagorean Theorem: Recognize when you can use the Pythagorean theorem to find the lengths of unknown sides, especially in right triangles formed within the trapezoid.

Further Exploration

Want to dig deeper? Try these: Change the base lengths and solve the problem again. What happens if the trapezoid isn't isosceles? How does that change things? Explore the relationship between the radius of the inscribed circle and the area of the trapezoid. Keep experimenting and have fun with math! You will be a geometry expert in no time!