In The Isosceles Trapezoid ABCD With AB Parallel To CD, Angle A = Angle B = 60°, And Diagonal AC Perpendicular To BC, Given BC = 12 Cm, Find: A) The Perimeter Of The Trapezoid; B) The Lengths Of Its Diagonals AC And BD.

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This article delves into the fascinating world of geometry, specifically focusing on the properties and calculations related to an isosceles trapezoid. We will explore a detailed solution to a problem involving trapezoid ABCD, where AB is parallel to CD, angles A and B are equal to 60 degrees, and diagonal AC is perpendicular to BC. Given that BC = 12 cm, we aim to determine the perimeter of the trapezoid and the lengths of its diagonals, AC and BD.

Understanding Isosceles Trapezoids

Before diving into the solution, let's establish a solid understanding of isosceles trapezoids. An isosceles trapezoid is a quadrilateral with one pair of parallel sides (bases) and the non-parallel sides (legs) being of equal length. This unique characteristic leads to several important properties:

  • The base angles are congruent (equal). In our case, angle A = angle B.
  • The diagonals are congruent. This means AC = BD.
  • The altitude (height) drawn from the endpoints of the shorter base to the longer base are congruent.

These properties will be instrumental in solving the problem at hand.

Problem Statement

We are given an isosceles trapezoid ABCD, where:

  • AB || CD (AB is parallel to CD)
  • ∠A = ∠B = 60°
  • AC ⊥ BC (AC is perpendicular to BC)
  • BC = 12 cm

Our goal is to find:

  • a) The perimeter of trapezoid ABCD
  • b) The lengths of diagonals AC and BD

Solution

a) Finding the Perimeter

To determine the perimeter, we need to find the lengths of all four sides of the trapezoid: AB, BC, CD, and DA. We already know BC = 12 cm. Since ABCD is an isosceles trapezoid, DA = BC = 12 cm.

Let's focus on finding AB and CD. Since AC ⊥ BC, triangle ABC is a right-angled triangle. We can use trigonometric ratios to find AB and AC.

Analyzing Triangle ABC

In right-angled triangle ABC, we have:

  • ∠B = 60°
  • BC = 12 cm

We can use the tangent function to find AC:

tan(60°) = AC / BC
√3 = AC / 12
AC = 12√3 cm

Now, we can use the cosine function to find AB:

cos(60°) = BC / AB
1/2 = 12 / AB
AB = 24 cm

Finding CD

To find CD, we need to draw altitudes from C and D to AB, let's call the points of intersection E and F respectively. This will create two congruent right-angled triangles, ADF and BCE, and a rectangle, CDEF.

  • AF = BE (due to the symmetry of the isosceles trapezoid)
  • CD = EF

In right-angled triangle BCE, we already know BC = 12 cm and ∠B = 60°. We can use the sine function to find CE (the height of the trapezoid):

sin(60°) = CE / BC
√3/2 = CE / 12
CE = 6√3 cm

Now, we can use the cosine function to find BE:

cos(60°) = BE / BC
1/2 = BE / 12
BE = 6 cm

Since AF = BE, AF = 6 cm. Also, because CDEF is a rectangle, CD = EF. We know that AB = AF + FE + EB, so:

24 = 6 + CD + 6
CD = 24 - 12
CD = 12 cm

Calculating the Perimeter

Now that we have all the sides, we can calculate the perimeter:

Perimeter = AB + BC + CD + DA

Perimeter = 24 cm + 12 cm + 12 cm + 12 cm

Perimeter = 60 cm

b) Finding the Lengths of the Diagonals

We already found the length of diagonal AC when analyzing triangle ABC:

AC = 12√3 cm

Since ABCD is an isosceles trapezoid, the diagonals are congruent. Therefore:

BD = AC = 12√3 cm

Summary of Results

  • a) The perimeter of trapezoid ABCD is 60 cm. This was calculated by finding the lengths of all four sides and summing them up. Understanding the properties of isosceles trapezoids and using trigonometric ratios were key steps in this calculation.
  • b) The lengths of the diagonals are AC = BD = 12√3 cm. We determined the length of AC using trigonometric ratios in the right-angled triangle ABC. Then, we applied the property that diagonals of an isosceles trapezoid are equal in length to find BD. This highlights the importance of remembering the properties of geometric shapes.

Key Concepts and Takeaways

  • Isosceles Trapezoid Properties: The congruent base angles and diagonals are critical for solving problems related to isosceles trapezoids. These properties allow us to make deductions and apply appropriate theorems and formulas. Recognizing these properties is often the first step in solving a geometric problem.
  • Trigonometric Ratios: Sine, cosine, and tangent are essential tools for working with right-angled triangles. By applying these ratios, we can relate angles and side lengths, allowing us to calculate unknown quantities. Practice using trigonometric ratios in various contexts to build proficiency.
  • Geometric Decomposition: Breaking down complex shapes into simpler ones (like triangles and rectangles) is a powerful problem-solving technique. This allows us to apply known formulas and theorems to smaller, more manageable parts. Learning to decompose shapes effectively is a crucial skill in geometry.
  • Problem-Solving Strategy: A structured approach to problem-solving involves understanding the given information, identifying the goal, choosing appropriate strategies, and executing the solution. This methodical approach helps ensure accuracy and efficiency. Developing a strong problem-solving strategy is invaluable in mathematics and beyond.

Conclusion

This problem demonstrates how a combination of geometric properties, trigonometric ratios, and problem-solving techniques can be used to find the perimeter and diagonal lengths of an isosceles trapezoid. By carefully analyzing the given information and applying relevant concepts, we successfully solved the problem. Geometry, at its core, is about understanding spatial relationships and using logic to derive new information. The more you practice, the better you will become at visualizing and solving geometric problems. This exercise showcases the beauty and power of geometric reasoning in action. Remember, every problem solved is a step towards mastering the art of geometry.