Test: Area Theorems - Class 9 MCQ

The areas of the triangles formed by connecting point P to the vertices of the parallelogram depend on the position of point P. If P is closer to one pair of sides, the areas of the triangles will differ according to their respective bases and heights.

The area of triangle DEF can be calculated as follows: Area = 1/2 × base × height = 1/2 × 12 × 5 = 30 cm².

When two triangles share the same base and have equal areas, their heights must also be equal. This is a direct result of the area formula, which shows that if the area remains constant and the base is unchanged, the height must also remain constant.

The area of the triangle is half that of the parallelogram when they share the same base and height. Therefore, if the area of the parallelogram is 60 cm², the area of the triangle is 1/2 × 60 = 30 cm².

Congruent figures have the same area and the same shape, meaning they are exactly identical in size and form. However, figures with equal areas may not be congruent if they are of different shapes or sizes.

A triangle can have the same area as a parallelogram if they share the same base and height, as the triangle's area is always half of that of the parallelogram based on the geometric properties of these shapes.

The areas of triangles that share a common vertex and have bases along the same line are proportional to the lengths of their respective bases. This relationship emphasizes how the base length influences the area of the triangle.

The corollary indicates that a parallelogram and a rectangle on the same base and between the same parallels have equal areas. This is important as it establishes a relationship between these two types of quadrilaterals under specific conditions.

The theorem states that triangles on the same base and between the same parallels have equal areas. This conclusion arises from the fact that both triangles share identical base lengths and heights, leading to equal area calculations.

The areas of the triangles formed by the diagonals of a quadrilateral can be compared using the ratios of their bases. This is due to the properties of triangle areas depending on their respective bases and heights relative to the diagonals.

The area of the triangle can be calculated using the formula Area = 1/2 × base × height. Plugging in the values, we get Area = 1/2 × 8 × 5 = 20 cm².

When two triangles have the same base and height, their areas are equal. This is because area is directly proportional to both base and height, so if these dimensions are identical, the resulting areas must also be the same.

The areas of the triangles formed by drawing segments from a point inside a parallelogram to its vertices can differ based on the location of that point. This illustrates how the position of a point affects area calculations in geometric figures.

If two triangles share the same base and have equal areas, their heights must also be equal. This is because the area is dependent on both the base and height; thus, equal areas imply equal heights when the base is constant.

The area of a triangle is half that of a parallelogram on the same base and between the same parallels. This relationship highlights the geometric principle that a triangle occupies half the area of a parallelogram when both share the same base and height.

Using the formula for the area of a triangle, Area = 1/2 × base × height, we find Area = 1/2 × 10 × 4 = 20 cm².

Theorem 19 states that parallelograms sharing the same base and lying between the same parallel lines have equal areas. This theorem can be proven using the properties of congruent triangles formed within the parallelograms.

If point D divides side BC in a ratio of 1:2, then triangle ABD will have an area that is 1/3 of triangle ABC, while triangle ACD will have an area of 2/3 of triangle ABC. This is due to the proportionality of the bases, which directly affects the area.

The area of a triangle is calculated using the formula Area = 1/2 × base × height. This formula reflects that the area of a triangle is half that of a rectangle with the same base and height, which provides a geometric visualization of the concept.

The area ratio of triangles ABD and ADC is 1:4 because the areas of the triangles are proportional to the lengths of their respective bases. Since D divides BC in a ratio of 1:4, the area of triangle ABD is one part while the area of triangle ADC is four parts.