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In △ ABC and △ ACD, ∠ ACB = ∠ ADC = 90 °, ab = 15cm, AC = 12cm. If the two right triangles are similar, then ad =? There seem to be two possibilities,
Because triangle ABC is similar to triangle ACD, the corresponding edge is proportional,
The first case: AD / AC = AC / AB, ad = acxac / AB = 12x12 / 15 = 9.6
The second case: AD / BC = AC / AB, so ad = acxbc / AB = 12x9 / 15 = 7.2
As shown in the figure, the edge ad of RT △ ACD
As shown in the figure, draw semicircles with the diameters of the sides ad, AC and CD of the isosceles right triangle ACD, and verify that the sum of the areas of the two crescent patterns AgCe and dhcf (the shaded part in the figure) is equal to the area of the RT triangle ACD
Proof
Let ad = 2R
∵△ ACD is a right angled RT triangle
∴AC=CD=√2R
Draw a semicircle with the diameter of AD, AC and CD
The ace area of semicircle = the CDF area of semicircle = 1 / 2 * π * (√ 2R / 2) & # 178; = π R & # 178 / 4
ACD area of semicircle = 1 / 2 * π * (2R / 2) & # 178; = π R & # 178 / 2
RT △ ACD area = 1 / 2 * ac * CD = R & # 178;
Area of arcuate ACG + area of arcuate CDH = area of semicircle ACD - RT △ ACD = π R & # 178; = (π / 2-1) R & # 178;
｜ shadow area = semicircular ace area + semicircular CDF area - (arcuate ACG area + arcuate CDH area)
=2*πR²/4-(π/2-1)R²
=R²
=RT △ ACD area
Known: as shown in the figure, in the quadrilateral ABCD, ∠ C = 60 °, DAB = 135 °, BC = 8, ab = 26, find the length of DC
As shown in the figure, be ∥ ad is crossed to CD through B, AF ⊥ be through a is crossed to CD through e, f ∥ BEC = ∥ ADC = 90 °, Abe = 180 ° - ∥ a = 45 °, AF = De, in RT △ BEC, CE = BC ∥ cos ∥ C = 8 × 12 = 4rt ⊥ ABF, AF = ab ∥ sin ∥ ABF = 26 × 22 = 23, ∥ DC = 4 + 23
Quadrilateral ABCD, connect AC, BD, angle bad is 30 degrees, triangle BCD is equilateral triangle, question: can line AB, ad, AC form right triangle, prove
Answer: can form right triangle
It is proved that △ ACD is rotated 60 degrees anticlockwise around point C to obtain △ ECB
So ∠ ace = 60 degrees, and AC = EC
So △ ace is an equilateral triangle
AE = AC
Because ∠ ADC + ∠ ABC = 360 - 30 - 60 = 270 degrees
Therefore, EBC + ABC = 270 degrees
Therefore, Abe = 90 degrees
So in the right angle △ Abe, there is ab ^ 2 + be ^ 2 = AE ^ 2
So AB ^ 2 + be ^ 2 = AC ^ 2
Be = ad
So AB ^ 2 + ad ^ 2 = AC ^ 2
So the lines AB, ad and AC can form a right triangle
(Note: due to limited conditions, in the answer, for example, AB ^ 2 represents the square of AB)
RELATED INFORMATIONS
- 1. It is known that the midpoint of BC is e and the midpoint of ad is f in the regular tetrahedron ABCD. The relationship between AE and CF is judged by AE and CF 1 It is known that in the regular tetrahedron ABCD, the midpoint of BC is e, the midpoint of ad is f, even AE and CF 1. Judge the relationship between AE and CF The cosine of the angle of 2 AE, CF
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- 9. It is known that there are four ABCD points on the sphere with radius 2. If AB = CD = 2, the maximum ABCD volume of the tetrahedron is? 3Q Such as the title
- 10. In space quadrilateral ABCD, if vector AB = (- 3,5,2), vector CD = (- 7, - 1, - 4), points E and F are the midpoint of edge BC and ad respectively, and vector EF is equal to?
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- 12. As shown in the figure, take the high ad on the hypotenuse BC of the isosceles right triangle ABC as the crease to make the planes of △ abd and △ ACD perpendicular to each other, The midpoint of BC is n (1) verification: plane ACD vertical plane BDC (2) verification: plane adn vertical plane ABC (3) degree of angle cab
- 13. Isosceles right triangle ABC, angle c is 90 degrees, D is a point on the side of BC, through B make the vertical line of ad at point E. find the angle AEC Sorry, I haven't learned the circle yet
- 14. As shown in the figure, in the right triangle ABC, ∠ C = 90 ° and BC is bisected by points D and E, and BC = 3aC, then ∠ AEC + ∠ ADC + ∠ ABC=______ °．
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- 16. As shown in the figure, in rectangular ABCD, diagonal lines AC and BD intersect at point O, ∠ BOC = 120 ° AB = 6, find: (1) the length of diagonal line; (2) the length of BC; (3) the area of rectangular ABCD
- 17. In rectangle ABCD, M is the midpoint of BC, Ma ⊥ MD. if the perimeter of rectangle is 48CM, the area of rectangle ABCD is______ cm2．
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- 19. The two diagonals of rectangle ABCD intersect at point O. given that the angle AOD is equal to 120 ° and ab is equal to 2.5cm, find the length of the diagonal of rectangle?
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