(1) If AB = 10, the distance between AB and CD is 8, AE = EB, BF = FC, find the area of △DEF.
(2) If the areas of △ADE, △BEF and △CDF are 5, 3 and 4 respectively, find the area of △DEF.
2. As shown in Figure 2, the square AEFG is obtained after the square ABCD rotates n counterclockwise around point A, and the side EF and CD intersect at point O. 。
(1) Connect two line segments (except the diagonal of a square) with the point marked with letters as the end point. It is required that the connected two line segments intersect and are perpendicular to each other, and explain the reasons why the two line segments are perpendicular to each other;
(2) If the side length of a square is 2cm and the area of the overlapping part (quadrilateral AEOD) is 4√3/3cm2, find the rotation angle n..
3. As shown in Figure 3, there is a circular iron sheet with a diameter of 2m. Make a covered oil drum, and try to use this iron sheet. The master intercepted two circles (i.e. two bottoms) and a rectangle (edge) on a circular iron sheet.
(1) If BC is taken as the height of the oil drum, what is the bottom radius R 1 of the oil drum?
(2) When AB is taken as the height of the oil drum, is the bottom radius R2 of the oil drum equal to r 1 in (1)? If they are equal, please explain the reasons; If not, request R2.
4. As shown in Figure 4, in trapezoidal ABCD, AD‖BC, AC⊥DB, AC = 5, ∠ DBC = 30 & ordm; .
(1) Find the length of diagonal BD;
(2) Find the area of trapezoidal ABCD.
5. As shown in Figure 5, E is the moving point on the AD side of the square ABCD, F is the point on the extension line of the BC side, BF = EF, AB = 12, and AE = X and BF = Y. 。
(1) When △BEF is an equilateral triangle, find the length of BF;
(2) Find the functional relationship between Y and X, and write the range of the independent variable X;
(3) Fold △ABE along the straight line BE, and point A falls on the point? Try to explore: △? Can it be an isosceles triangle? If yes, request the length of AE; If not, please explain why.
6. As shown in Figure 6, P is a point in the square ABCD, and PA∶PB∶PC= 1∶2∶3. Find the degree of ∠APB.
7. As shown in Figure 7, it is known that the two diagonals of diamond ABCD are A and B respectively. Take each side as the diameter, make a semicircle into a diamond shape, and find the petal-like area surrounded by four semicircles (that is, the area of the shaded part in the figure).
8. As shown in Figure 8, in the plane rectangular coordinate system, OEFG is a square, and the coordinates of point F are (1, 1). Place the right vertex of a right-angled triangular paper with the shortest side longer than √2 on the diagonal FO.
(1) When the right-angle vertex of a triangular piece of paper coincides with point F, and a right-angle edge falls on the straight line FO, it is easy to know that the area of the overlapping part of the triangular piece of paper and the square OEFG (that is, the shadow part) is1/2;
(2) If the right-angle vertex of triangular paper does not coincide with points O and F, and two right-angle sides intersect with two adjacent sides of the square (that is, the right-angle vertex of triangular paper slides on the diagonal FO of the square), when the overlapping area of triangular paper and square OEFG is half of the square area, try to find the coordinates of the right-angle vertex of triangular paper and draw it.