In the parallelogram ABCD, am ⊥ BC, an ⊥ CD, m and N are perpendicular feet. If AB = 13, BM = 5 and MC = 9, the length of Mn is______ ．

In ∵ am ⊥ BC, ab = 13, BM = 5, ∵ RT △ ABM, am = AB2 − BM2 = 12, SINB = amab = 1213, ∵ quadrilateral ABCD is parallelogram, ∵ d = ∵ B, ∵ an ⊥ CD, ad = BC = BM + MC = 14, ∵ in RT ⊥ adn, sind = anad = 1213, ∵ an = 16813, ∵ amn, am = 12, ∵ B + ∠ ba

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1. In the parallelogram ABCD, am is perpendicular to BC and an is perpendicular to CD. It is proved that am: ab = Mn: AC
2. As shown in the figure, square ABCD, M is a point on BC. Connect am, make the vertical bisector GH of am, intersect AB at point G, intersect CD at point h. given am = 10cm, find the length of GH
3. As shown in the figure, square ABCD, M is a point on BC. Connect am, make the vertical bisector GH of am, intersect AB at point G, intersect CD at point h. given am = 10cm, find the length of GH
4. As shown in the figure, in the square ABCD with side length of 8, M is a point on BC. Connect am, make the vertical bisector GH of am, intersect AB with G, intersect CD with H. if cm = 2, then GH=______ ．
5. As shown in the figure, square ABCD, M is a point on BC. Connect am, make the vertical bisector GH of am, intersect AB at point G, intersect CD at point h. given am = 10cm, find the length of GH
6. As shown in the figure, in the square ABCD with side length of 8, M is a point on BC. Connect am, make the vertical bisector GH of am, intersect AB with G, intersect CD with H. if cm = 2, then GH=______ ．
7. In square ABCD, am intersects BC at m, an intersects DC at n. angle man is 45 degrees. Angle man rotates clockwise around a, and its two sides intersect CB and DC at m respectively In square ABCD, am intersects BC at point m, an intersects DC at point n. angle man is 45 degrees. Angle man rotates clockwise around point a, and its two sides intersect CB and DC at point m and N respectively. When angle man rotates around point a to BM = DN, it is easy to prove BM + DN = Mn. When angle man rotates around point a to BM not = DN, what is the relationship between them?
8. It is known that in square ABCD, ∠ man = 45 ° and ∠ man rotates clockwise around point a, and its two sides intersect CB and DC (or their extension lines) at points m and N respectively. When ∠ man rotates around point a to BM = DN (as shown in Figure 1), it is easy to prove that BM + DN = Mn (1) When ∠ man rotates around point a to BM ≠ DN (as shown in Figure 2), what is the quantitative relationship among BM, DN and Mn? Write out the conjecture and prove it; (2) when ∠ man rotates around point a to the position as shown in Fig. 3, what is the quantitative relationship among line segments BM, DN and Mn? Please write your guess directly
9. As shown in the figure, in rectangle ABCD, ab = 16, BC = 8, fold the rectangle along AC, point d falls at point E, and CE and ab intersect at F, then AF=______ ．
10. As shown in the figure, in rectangle ABCD, ab = 16, BC = 8, fold the rectangle along AC, point d falls at point E, CE and ab intersect at point F, (1) prove: AF = CF (2) find the length of AF
11. Parallelogram ABCD, ab = BC, ∠ B = 60 °, ACD = 60 °, M is the point on BC, n is the point on CD, ∠ amn = 60 ° asks the relation between AM and Mn
12. M. N are the points on the sides of BC and CD of parallelogram ABCD respectively, and Mn is parallel to BD. it is proved that s triangle adn = s triangle ABM
13. In the parallelogram ABCD, am ⊥ BC, an ⊥ CD, m and N are perpendicular feet. If AB = 13, BM = 5 and MC = 9, the length of Mn is______ ．
14. Through the vertex a of parallelogram ABCD, make am ⊥ BC, an ⊥ CD, m and N are perpendicular feet. If ∠ man = 30 °, CD = 4cm, BC = 6cm, then am =? An =?
15. As shown in the figure, the points m and N are on the sides BC and CD of the square ABCD respectively. It is known that the perimeter of △ MCN is equal to half of the perimeter of the square ABCD, then ∠ man=______ ．
16. The height of the BC side of the parallelogram ABCD is 12 cm, and the height of the CD side is 15 cm. If the perimeter of the parallelogram ABCD is 72 cm, what is the area of the parallelogram in square cm?
17. It is known that in the parallelogram ABCD, the diagonal lines AC and BD intersect at O, m and N, which are the midpoint of OA and OC respectively. What is the relationship between BM and DN?
18. As shown in the figure, in the parallelogram ABCD, AC and BD intersect at point O, and m and N are the midpoint of OA and OC respectively
19. In the triangle ABC, D and E are the points on the sides of AB and AC respectively, and De is parallel to BC, be and DC intersect at O, and the line AO and BC intersect at M
20. It is known that in the triangle ABC, D and E are points on AB and AC respectively, and de / / BC, be and CD intersect at point O, and the extension line of Ao intersects with BC at point M=