Question

In-a trapezium \(A B C D\), the diagonals \(A C\) and \(B D\) intersectes other.at \(O\) such that \(O B: O D=3: 1\) then the ratio of aresj \(\triangle A O B: \triangle C O D\) is : (a) \(3.1\) (c) \(9: 1\) (b) \(1: 4\) (d) can't be determined

Short answer

Answer: The ratio of the areas of triangles AOB and COD is 3:1.

Step-by-step solution

Understand the given information and draw a diagram

To better visualize the problem, draw a diagram of the trapezium ABCD, with diagonals AC and BD intersecting at O, and OB:OD being 3:1.

Divide the trapezium into two triangles

Divide trapezium ABCD into two triangles by drawing a line segment from vertex A to vertex C. Now we have two triangles: AOB and COD.

Use the property of triangles

We can use the property that the ratio of the areas of two triangles with the same height is equal to the ratio of their bases. Let h be the height of both triangles AOB and COD.

Calculate the ratio of the bases

Since OB: OD is 3:1, let OB = 3x, and OD = x. The base of triangle AOB is 3x and the base of triangle COD is x. Thus, the ratio of the bases is 3x:x = 3:1.

Calculate the ratio of the areas of triangles AOB and COD

We know that the ratio of the areas of triangles AOB and COD is equal to the ratio of their bases, which is 3:1. So, Area(AOB):Area(COD) = 3:1.

Choose the correct answer

The ratio of the areas of triangles AOB and COD is 3:1. This corresponds to option (a) \(3:1\) in the problem statement. Therefore, the correct answer is (a) \(3:1\).

Key concepts

Triangle Area Properties

One of the fundamental properties of triangles is the relationship between the base, the height, and the area. The area of any triangle can be calculated using the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
This equation shows that if two triangles have the same height, the ratio of their areas will be directly proportional to the ratio of their bases. This is a key concept when comparing areas within geometric figures, like the trapezium in our exercise.

In the context of the exercise, we saw the trapezium being divided into two triangles, \(\triangle AOB\) and \(\triangle COD\), by the diagonal \(AC\). Both triangles share the same height, i.e., the perpendicular distance between the line \(AC\) and the base \(BD\). Therefore, their areas are influenced only by their respective bases, which in this case are the line segments \(OB\) and \(OD\). As we calculated the ratio of their areas using the given ratio of \(OB:OD = 3:1\), we could confidently determine the relative sizes of their areas based purely on triangle area properties.

Ratio of Areas in Geometry

The concept of the ratio of areas is essential in geometry. It allows us to compare different sections of a shape without the need to calculate the exact measurement of each area. Ratios depict a relative size, serving as a tool to deduce one measurement from another.

When given a ratio of sides, like \(OB:OD\) in the trapezium exercise, we applied this ratio to understand the relationship between the areas of \(\triangle AOB\) and \(\triangle COD\). This is particularly useful in geometric figures where exact measurements are not known, but the proportional relationship between sides can lead to conclusions about the area relationships.

Focusing on the core geometric principle that triangles with equivalent heights will have areas proportional to their bases, we are empowered to extend this principle to complex figures. In essence, by breaking down a polygon into simpler shapes such as triangles, we can analyze the polygon's properties through the lens of these simpler shapes.

Trapezium and Triangles

A trapezium, known as a trapezoid in some regions, is a four-sided figure with at least one pair of parallel sides. When we examine a trapezium through the lens of its triangular sections, as we've done with the diagonals \(AC\) and \(BD\), we gain insight into the trapezium's properties. Diagonals in a trapezium typically create two distinct triangles which often share common features like a height or angle.

In the given exercise, when the trapezium \(ABCD\) is divided by its diagonals, we observed how their point of intersection, \(O\), established a ratio between \(OB\) and \(OD\). Understanding that the areas of \(\triangle AOB\) and \(\triangle COD\) are proportional to these segments allowed us to determine the ratio of their areas. This approach to breaking down complex figures into triangles can simplify many problems in geometry, aiding in the understanding of how larger, more complicated shapes, like trapeziums, behave.