Introduction to Geometry Problems and Their Solutions

Geometry often involves solving for areas, volumes, angles, and lengths of various figures. One such problem is calculating the area of a trapezium given its side lengths. This guide will walk you through solving this problem step-by-step, providing a clear and detailed explanation along with the necessary calculations.

Problem Statement

The parallel sides of a trapezium are 12 meters and 26 meters, and the lengths of the non-parallel sides are 13 meters and 15 meters. The objective is to determine the area of this trapezium.

Approach to Solving the Problem

To find the area of the trapezium with the given side lengths, we first need to check if it can be considered a cyclic quadrilateral, which would allow us to use Brahmagupta's formula for the area. However, this approach won't work in this case. Instead, we will use an alternative method based on the Pythagorean theorem to find the height of the trapezium and then use the area formula for a trapezium.

Step 1: Check for Cyclic Trapezium

To determine if the trapezium is cyclic, we need to check if the sum of the lengths of the opposite sides are equal. However, in this case, the trapezium is not cyclic because:

12 26 ! 13 15

Hence, we cannot directly use Brahmagupta's formula.

Calculating the Area Using the Height

The area of a trapezium can be found using the formula:

A 1/2 × (a b) × h

where 'a' and 'b' are the lengths of the parallel sides, and 'h' is the height.

Step 2: Finding the Height Using the Pythagorean Theorem

Let's drop perpendiculars from the endpoints of the shorter base to the longer base, forming two right triangles. The length of the segments on the base 'b' created by these perpendiculars will help us find the height 'h'.

Step 3: Use the Pythagorean Theorem

Let 'x' be the distance from the left endpoint of the shorter base 'a' to the foot of the perpendicular dropped from the left endpoint of the longer base 'b'. The remaining length on the base 'b' will be 26 - x.

Using the Pythagorean theorem for the two triangles formed:

Left triangle: h2 x2 132 ≈ 169 Right triangle: h2 (26 - x)2 152 ≈ 225

Step 4: Solve for x and h

Expanding and equating the equations:

h2 x2 169

h2 676 - 52x x2 225

Merging terms and simplifying:

h2 x2 - 52x 676 225

h2 - 52x 451 0

Substituting h2 169 - x2 into the equation:

169 - x2 - 52x 451 0

620 - 52x 0

52x 620

x ≈ 11.9231

Now, substituting x back into the equation for h2:

h2 169 - (11.9231)2

h2 ≈ 169 - 142.2521 ≈ 26.7479

Therefore, h ≈ √26.7479 ≈ 5.174 meters

Step 5: Calculate the Area

Substituting the values back into the area formula:

A 1/2 × (12 26) × 5.174

A 1/2 × 38 × 5.174 ≈ 98.316 square meters

Final Answer: The area of the trapezium is approximately 98.32 square meters.