Question

In a trapezium \(A B C D, A B\) is a parallel to \(C D, B D\) is perpendicular to AD. \(A C\) is perpendicular to \(B C\). If \(A D=B C=15 \mathrm{~cm}\) and \(A B=25 \mathrm{~cm}\), then the area of the trapezium is : (a) \(192 \mathrm{~cm}^{2}\) (b) \(232 \mathrm{~cm}^{2}\) (c) \(172 \mathrm{~cm}^{2}\) (d) none of these

Short answer

(a) 192 cm^2 (b) 234 cm^2 (c) 312 cm^2 (d) 125 cm^2

Step-by-step solution

Find side CD

Since AB is parallel to CD, and AC is perpendicular to BC, we have a right triangle ADC, with AD = 15 cm and AC = height. Also, since AB = 25 cm, CD can be found as follows: CD = AB - AD = 25 - 15 = 10 cm.

Find the height

Using the Pythagorean theorem, we can find the height by solving the right triangle ADC: height^2 = AD^2 - CD^2 = 15^2 - 10^2 = 225 - 100 = 125. Taking the square root of both sides, we get: height = sqrt(125) = 5 * sqrt(5) cm.

Calculate the area

Now, we can use the formula for the area of a trapezium to find the area: Area = (1/2) * (sum of parallel sides) * height Area = (1/2) * (AB + CD) * height Area = (1/2) * (25 + 10) * 5 * sqrt(5) Area = (1/2) * 35 * 5 * sqrt(5) Simplify the expression to get the exact area. Area = 87.5 * sqrt(5) Now, we can approximate the area as sqrt(5) is approximately 2.236. Area ≈ 87.5 * 2.236 = 195.65 cm^2 Since this number is closest to 192 cm^2, we can choose the answer (a). Final answer: (a) \(192 \mathrm{~cm}^{2}\)

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry, particularly useful when dealing with right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship can be represented by the equation: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

For students tackling trapezium area calculation challenges, the Pythagorean theorem can be applied to find the height of the trapezium when one side is perpendicular to another, as based on the original problem provided. By rearranging the formula to solve for the unknown side, and then taking the square root, we can find the value needed to compute the area of a trapezium, as seen in the step-by-step solution.

Geometric Shapes

In geometry, a variety of shapes and figures are studied, each with its own set of properties and formulas for calculation. Among these shapes, trapeziums (or trapezoids, as they are known in some regions) have a distinctive feature: they are four-sided figures with at least one pair of opposite sides that are parallel.

Understanding the properties of geometric shapes like trapeziums is crucial for solving related problems. In the trapezium, the parallel sides are known as bases, and the distance between them, which is perpendicular, is called the height or altitude. For the area calculation, a trapezium can be seen as a combination of triangles and rectangles, simplifying the approach to find its measure. Knowing the bases and the height enables the use of the area formula for a trapezium, which integrates these dimensions.

Triangle Properties

Triangles are three-sided polygons that come in various forms, such as equilateral, isosceles, and scalene, and they can be classified by their angles as well. For calculations in geometry, knowing the properties of triangles is invaluable. Specifically, right-angled triangles, where one angle is exactly 90 degrees, are frequently used in conjunction with the Pythagorean theorem.

In the context of our trapezium problem, the right-angled triangles created by the perpendicular sides are utilized to derive necessary lengths for the area calculation. The properties of these triangles allow us to apply formulas such as the Pythagorean theorem to find missing dimensions, which then feed into the formula for the trapezium's area. Remember that in a right triangle, the side opposite the right angle is always the longest side and is used as the hypotenuse in the Pythagorean theorem.