Area Of Parallelogram And Triangle

Explanation
If a triangle and square are on the same base and between the same parallels, their heights will be the same. The area of a triangle is calculated by multiplying its base by its height and dividing the result by 2. The area of a square is calculated by multiplying its side length by itself. Since the heights are the same, the ratio of the area of the triangle to the area of the square will be equal to the ratio of the base of the triangle to the side length of the square, which is 1:2.

Explanation
The area of a parallelogram is calculated using the formula Area=Base × Height. This formula requires knowing the length of the base and the vertical height (the perpendicular distance) from the base to the opposite side of the parallelogram. Unlike a triangle, the area is not halved. The product of the base and the height gives the total square units inside the parallelogram, reflecting its entire two-dimensional extent.

Explanation
When we join the midpoints of the adjacent sides of a rectangle, we create a diagonal that divides the rectangle into two congruent triangles. The diagonal also bisects each of the four sides of the rectangle. Since the rectangle has sides of 8 cm and 6 cm, the diagonal has a length of 10 cm (by using the Pythagorean theorem). The diagonal of a rhombus bisects its angles, so when we draw the diagonal of the rhombus, it creates four congruent right triangles. Each of these triangles has a base of 4 cm (half of the length of the rectangle's side) and a height of 3 cm (half of the length of the rectangle's other side). Therefore, the area of each triangle is (1/2) x 4 cm x 3 cm = 6 cm². Since there are four congruent triangles, the total area of the rhombus is 4 x 6 cm² = 24 cm². Therefore, the correct answer is a rhombus of area 24 cm².

Explanation
In a triangle, when the midpoints of the sides are connected, the resulting triangle is called the medial triangle. The medial triangle is always similar to the original triangle and its area is always 1/4th of the area of the original triangle. Therefore, the ratio of the area of triangle DEF to triangle ABC is 1:4.

Explanation
The area of a parallelogram is calculated by multiplying the base length by the altitude. In this case, the base is 8 cm and the altitude is 5 cm. Multiplying these values together gives us an area of 40 cm2.

Explanation
The area of the parallelogram ABCD can be found by doubling the area of the triangle EFG. Since E, F, G, and H are the midpoints of the sides of the parallelogram, triangle EFG is half the size of the parallelogram. Therefore, if the area of triangle EFG is 40 cm2, the area of the parallelogram ABCD is 2 times that, which is 80 cm2.

Explanation
The area of a square can be found by multiplying the length of one side by itself. In this case, we are given the length of the diagonal, which we can use to find the length of one side using the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. So, if the length of the diagonal is 10 cm, then the length of one side is 10/√2 cm. Therefore, the area of the square is (10/√2) * (10/√2) = 100/2 = 50 cm2.

Explanation
In a parallelogram, the diagonals bisect each other. Therefore, diagonal AC divides parallelogram ABCD into two congruent triangles, CAB and CDA. Since E and F are the midpoints of AB and CD respectively, they divide the diagonals in half. Thus, triangle CEF is also congruent to triangle CAB. As a result, the ratio of the areas of triangle CEF to triangle CAB is 1:1, making the answer 1:1.

Explanation
The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides. In this case, the trapezium ABCD can be divided into two triangles, ABF and CDE, with the same height. The ratio of their bases is (3a + b):(a + 3b). Since the areas of triangles are directly proportional to their bases, the ratio of the areas of ABF and CDE is also (3a + b):(a + 3b). Therefore, the correct answer is (3a + b):(a + 3b).

Explanation
The centroid of a triangle divides each median into two segments, with the segment joining the centroid to the vertex being twice as long as the segment joining the centroid to the midpoint of the opposite side. Therefore, the area of triangle AGC is 1/3 of the area of triangle ABC.

Explanation
Since 'O' is the midpoint of AB, it divides the parallelogram into two congruent triangles, AOD and BOC. Therefore, the area of triangle DOC is half the area of the parallelogram. Given that the area of the parallelogram is 74 sq cm, the area of triangle DOC would be half of that, which is 37 sq cm.

Explanation
Since the parallelogram and rectangle have equal areas, it means that their bases (AB and EF) are equal. However, the height of the parallelogram is greater than the height of the rectangle, which means that the length of the sides (BC and AD) of the parallelogram are longer than the length of the sides (BE and AF) of the rectangle. Therefore, the perimeter of ABCD, which is the sum of the lengths of all four sides, will be greater than the perimeter of ABEM, which is the sum of the lengths of only three sides.

Explanation
If the diagonal AC of quadrilateral ABCD divides it into two parts equal in area, then it is not necessarily a rectangle because a rectangle requires all angles to be 90 degrees. It is also not necessarily a rhombus because a rhombus requires all sides to be equal in length. However, it is always a parallelogram because a parallelogram is a quadrilateral with opposite sides that are parallel, and the diagonal AC dividing it into equal areas implies that the opposite sides are equal in length.

Explanation
The median of a triangle divides it into two triangles of equal area. This means that if we draw a line segment from one vertex of the triangle to the midpoint of the opposite side, the triangle will be divided into two smaller triangles that have the same area. This property of the median is true for all triangles, regardless of their shape or size.