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It is known that: as shown in the figure, in △ ABC, ∠ BCA = 90 °, D and E are the midpoint of AC and ab respectively, and point F is on the extension line of BC, and ∠ CDF = ∠ A. (1) prove that the quadrilateral decf is a parallelogram; (2) bcab = 35, the perimeter of quadrilateral ebfd is 22, and calculate the area of quadrilateral decf. (Note: the middle line on the hypotenuse of a right triangle is equal to half of the hypotenuse.)
(1) It is proved that: ∵ AE = EB, ad = DC, ≌ ed ∥ BC. ∵ point F is on the extension line of BC, ∵ ed ∥ CF. ∵ ad = DC, ed = De, ≌ ade = ≌ EDC, ≌ ade ≌ CDE. ∥ a = ∥ ECD. ∥ CDF = ∥ a, ∥ CDF = ∥ ECD. ∥ EC ∥ DF. ∥ quadrilateral decf is a parallelogram. (2)
RELATED INFORMATIONS
- 1. If AB is 1:4 to de, the sum of the circumference of the equilateral triangle ABC and the square defg is 76 cm, what is the circumference of the triangle? AB is one side of an equilateral triangle, De is one side of a square,
- 2. The side length of equilateral triangle ABC is a, and the square defg is inscribed with △ ABC, D, E on AB, AC, G, f on BC respectively. Find the side length of square defg
- 3. As shown in the figure △ ABC, ∠ a = 90, de bisects AB vertically, intersects BC with EAB = 20, AC = 12, finds the length of be and the area of quadrilateral Adec
- 4. As shown in Figure 1, fold the △ ABC paper along De, so that point a falls at the position of point a 'in the quadrilateral bced. Through calculation, we know that: 2 ∠ a = ∠ 1 + ∠ 2 (1) If the △ ABC paper is folded along the de so that the point a falls on the position of the outer point a 'of the quadrilateral bced, as shown in Fig. 2, what is the relationship between ∠ A and ∠ 1, 2? Why? Please explain the reason. (2) if the quadrilateral ABCD is folded along EF so that the points a and d fall on the positions of a 'and d' inside the quadrilateral BCFE, as shown in Figure 3, can you find out the relationship between ∠ a, ∠ D, ∠ 1 and ∠ 2? (write the relation directly)
- 5. (1) As shown in Figure 1, fold the △ ABC paper along de so that point a falls at the position of point a 'inside the quadrilateral bced, and try to explain that 2 ∠ a = ∠ 1 + ∠ 2; (2) as shown in Figure 2, if the △ ABC paper is folded along de so that point a falls at the position of point a' outside the quadrilateral bced, then the equivalent relationship between ∠ A and ∠ 1, ∠ 2 is______ (no need to explain the reason); (3) as shown in Figure 3, if the quadrilateral ABCD is folded along EF, so that points a and d fall on the positions of internal points a ′, D ′ of quadrilateral BCFE, please explore the quantitative relationship between ∠ a, D, 1 and 2 at this time, write down the conclusion you found and explain the reason
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