Find The Perimeter Of Triangle ADC Given A Circle C(O; R) With Radius R = 6 Cm, Points A, B, C, And D Such That AB || CD, O ∈ AB, And The Measure Of Arc AD Is 60 Degrees.

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In the fascinating realm of geometry, circles hold a special allure. Their perfect symmetry and elegant properties have captivated mathematicians for centuries. Today, we embark on a journey to explore the geometry of a circle, specifically focusing on calculating the perimeter of triangle ADC within a circle 𝐶(𝑂; 𝑅) with radius R = 6 cm. Let's delve into the intricacies of this geometric puzzle, unraveling its secrets step by step.

Problem Statement

We are given a circle 𝐶(𝑂; 𝑅) with radius R = 6 cm. Within this circle, we have four points: A, B, C, and D. These points are strategically positioned such that 𝐴𝐵∥𝐶𝐷, the center of the circle O lies on the line segment AB, and the measure of arc AD is equal to 60°. Our mission is to determine the perimeter of triangle ADC.

Visualizing the Geometry

Before we dive into calculations, let's paint a mental picture of the scenario. Imagine a circle with its center O. Now, picture a line segment AB passing through O, effectively dividing the circle into two halves. Next, envision another line segment CD, parallel to AB, but not necessarily passing through the center. Finally, mark points A, D, and C on the circumference of the circle, forming triangle ADC. The arc AD, a portion of the circle's circumference, subtends an angle of 60° at the center O. This visual representation will serve as our compass as we navigate the geometric landscape.

Unlocking the Solution

Now, let's embark on the quest to calculate the perimeter of triangle ADC. To achieve this, we need to determine the lengths of the three sides: AD, DC, and AC. We will employ a blend of geometric principles, trigonometric relationships, and a touch of ingenuity to unveil these lengths.

1. Finding the Length of AD

Our journey begins with AD, the side subtended by the 60° arc. Recall that the length of an arc is proportional to the central angle it subtends. In this case, the central angle ∠AOD corresponds to the arc AD. The formula connecting arc length, radius, and central angle is:

Arc length = (Central angle / 360°) × 2πR

Plugging in the values, we get:

AD = (60° / 360°) × 2π(6 cm) = 2π cm

However, we need the length of the chord AD, not the arc length. To bridge this gap, we draw upon the properties of isosceles triangles. Triangle AOD is isosceles since OA = OD = R (radii of the circle). The central angle ∠AOD is 60°, making the other two angles equal: ∠OAD = ∠ODA = (180° - 60°) / 2 = 60°. This revelation transforms triangle AOD into an equilateral triangle, where all sides are equal. Therefore, AD = OA = OD = 6 cm.

2. Determining the Length of DC

Next, we set our sights on DC. Here, the parallelism between AB and CD comes into play. Since AB∥CD, the alternate interior angles are equal. Let's denote the intersection points of AD and BC with CD as E and F, respectively. Then, ∠ADE = ∠BAO and ∠BCF = ∠CBO. These angle equalities will prove crucial in our quest to find DC.

To proceed, we draw perpendiculars from O to CD, meeting CD at point M. This creates two right-angled triangles: ΔOMC and ΔOMD. Since O is the center of the circle, OM bisects CD, making CM = MD. Now, let's focus on triangle OMC. We know OC = R = 6 cm. To find OM, we need to consider the angle ∠COM. Since AB∥CD, ∠AOD and ∠COD are supplementary angles, meaning they add up to 180°. Thus, ∠COD = 180° - ∠AOD = 180° - 60° = 120°. Since OM bisects ∠COD, ∠COM = 120° / 2 = 60°.

Now, in right-angled triangle OMC, we have:

sin(∠COM) = CM / OC

sin(60°) = CM / 6 cm

CM = 6 cm × sin(60°) = 6 cm × (√3 / 2) = 3√3 cm

Since CM = MD, DC = 2 × CM = 2 × 3√3 cm = 6√3 cm.

3. Unveiling the Length of AC

Our final challenge is to determine the length of AC. This side connects points A and C, and its length requires a different approach. We will employ the Law of Cosines, a powerful tool that relates the sides and angles of a triangle.

Consider triangle AOC. We know OA = OC = R = 6 cm. The angle ∠AOC is the central angle subtended by arc AC. To find ∠AOC, we subtract ∠AOD from ∠COD: ∠AOC = ∠COD - ∠AOD = 120° - 60° = 60°.

Now, applying the Law of Cosines to triangle AOC:

AC² = OA² + OC² - 2 × OA × OC × cos(∠AOC)

AC² = (6 cm)² + (6 cm)² - 2 × 6 cm × 6 cm × cos(60°)

AC² = 36 cm² + 36 cm² - 72 cm² × (1/2)

AC² = 36 cm²

AC = √36 cm² = 6 cm

Calculating the Perimeter

With the lengths of all three sides in hand, we can now calculate the perimeter of triangle ADC:

Perimeter of ADC = AD + DC + AC

Perimeter of ADC = 6 cm + 6√3 cm + 6 cm

Perimeter of ADC = 12 cm + 6√3 cm

Therefore, the perimeter of triangle ADC is 12 + 6√3 cm.

Conclusion

Our geometric journey has led us to the successful calculation of the perimeter of triangle ADC. By skillfully employing geometric principles, trigonometric relationships, and the Law of Cosines, we have unraveled the lengths of the triangle's sides and determined its perimeter. This exploration serves as a testament to the elegance and power of geometry in solving intriguing problems.

Keywords

  • Perimeter of triangle ADC
  • Circle geometry
  • Arc length
  • Central angle
  • Law of Cosines
  • Isosceles triangle
  • Equilateral triangle
  • Parallel lines
  • Alternate interior angles
  • Trigonometric relationships

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Calculating the Perimeter of Triangle ADC in a Circle Geometry Problem