Vene T. Bogoslavov: Collection of Solved Problems #3
MR-679 / 1.)

Calculate the area of a rectangle with a perimeter of 14 dm and a diagonal of 5 dm.

MR-681 / 2.)

Calculate the area of the rectangle, if its sides are related as 3:4, and the radius of the circumscribed circle is 1 dm.

MR-683 / 3.)

In an acute triangle with a base of 10 cm and a height corresponding to the base of 8 cm, a rectangle is inscribed whose two vertices belong to the base of the triangle, and the other two on the other two sides. If the area of the rectangle is 15 cm2, calculate its sides.

MR-684 / 4.)

The sides of the rectangle are 3cm and 1cm. All four bisectors of its angles are constructed to their mutual intersections. Calculate the area of the resulting quadrilateral whose vertices are the intersecting points of the bisectors.

MR-685 / 5.)

In the isosceles right triangle ABC, the bisector of the acute angle B is constructed, which intersects the leg AC at the point M. Squares are constructed over the lengths AM and MC, as over the sides. Prove that the area of one square is twice the area of another.

MR-686 / 6.)

Calculate the area of a parallelogram whose heights are 3 cm and 2√3 cm, and the angle between them is 60°.

MR-688 / 7.)

The heights of the parallelogram are related as 2:3, its circumference is 40 cm, and the acute angle is 30°. Calculate the area of the parallelogram.

MR-687 / 8.)

The area of the parallelogram is 36 m2, and the distances between the intersecting point of the diagonals and the sides are 2 m and 3 m. Calculate the perimeter of the parallelogram.

MR-689 / 9.)

An isosceles trapezoid with a base of 8 cm and 2 cm is described around a circle. Calculate the area of the trapezoid.

MR-159 / 10.)

The area of an isosceles trapezoid circumscribed around a circle is 50 cm2, and the acute angle at the base is 30°. Determine the leg of the trapezoid.

MR-691 / 11.)

Calculate the area of a parallelogram whose diagonals are 26 cm and 30 cm, and whose side is 14 cm.

MR-692 / 12.)

The bases of the trapezoid are 142 cm and 89 cm, and the diagonals are 120 cm and 153 cm. Calculate the area of the trapezoid.

MR-694 / 13.)

The bases of the trapezoid are 24 cm and 10 cm, and the arms are 13 cm and 15 cm. Calculate the area of the trapezoid.

MR-695 / 14.)

The sides of the triangle are 13 cm, 14 cm and 15 cm. A line parallel to the largest side of the triangle cuts off a trapezoid with a perimeter of 39 cm. Calculate the area of the trapezoid.

MR-696 / 15.)

Calculate the area of an isosceles triangle with a base of 12 cm, if the height corresponding to the base is equal to the segment joining the midpoints of the base and the leg.