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As shown in the figure, △ ABC, ab = AC, points D and E are on the extension line of AB and AC respectively, and BD = CE, de and BC intersect at point F. verification: DF = EF
It is proved that DG ∥ AE is made through point D and passed to point BC at point G, as shown in the figure,
∴∠1=∠2，∠4=∠3，
∵AB=AC，
∴∠B=∠2，
∴∠B=∠1，
∴DB=DG，
And BD = CE,
∴DG=CE，
In △ DFG and △ EFC
∠4＝∠3
∠DFG＝∠EFC
DG＝CE ，
∴△DFG≌△EFC，
∴DF=EF．
As shown in the figure, △ ABC, ab = AC, points D and E are on the extension line of AB and AC respectively, and BD = CE, de and BC intersect at point F. verification: DF = EF
It is proved that DG ∥ AE is made through point D and passed to point BC at point G, as shown in the figure,
∴∠1=∠2，∠4=∠3，
∵AB=AC，
∴∠B=∠2，
∴∠B=∠1，
∴DB=DG，
And BD = CE,
∴DG=CE，
In △ DFG and △ EFC
∠4＝∠3
∠DFG＝∠EFC
DG＝CE ，
∴△DFG≌△EFC，
∴DF=EF．
As shown in the figure, AB is the diameter of ⊙ o, point C is on ⊙ o, bisector of ⊙ cab intersects ⊙ o at point D, passes through point D as the vertical line of AC, crosses the extension line of AC at point E, and connects BC with AD at point F (1) Guess the position relationship between ED and ⊙ o, and prove your conjecture; (2) If AB = 6, ad = 5, find the length of AF
(1) The relationship between ED and ⊙ o is tangent. The reasons are as follows: connect OD, ∵ cab bisector intersects ⊙ o at point D, ᙽ CD = BD, ⊥ OD ⊥ BC, ∵ AB is the diameter of ⊙ o, ∵ ACB = 90 °, that is BC ⊥ AC, ∵ de ⊥ AC,
As shown in the figure, in RT △ ABC, ∠ C = 90 °, ab = 50, AC = 30, D, e, f are the midpoint of AC, AB and BC respectively. Point P starts from point D and moves along the broken line de-ef-fc-cd at a speed of 7 units per second; point Q starts from point B at a speed of 4 units per second in the direction of Ba, and passes through point Q as the ray QK ⊥ AB, and the intersecting broken line bc-ca starts from point g. points P and Q at the same time, when point P makes a circle When point d stops moving, point Q also stops. Set point P and Q to move for T seconds (T > 0) (1) The distance between D and F is______ ; (2) Can X-ray QK divide a quadrilateral cdef into two parts of equal area? If it can, find the value of T; if not, explain the reason; (3) When the point P moves to the broken line ef-fc, and the point P falls on the ray QK, calculate the value of T; (4) Connect PG, when PG ∥ AB, write the value of t directly
(1) In RT △ ABC, ∠ C = 90 ° AB = 50, ∵ D, f is the midpoint of AC and BC, ᙽ DF is the median line of △ ABC,
In RT △ ABC, ∠ C = 90 ° if AC = 3, BC = 4, ab = 5, find the length of CD
The vertical line of AB line passing through point C? 3 * 4 / 5 = 2.4
In RT △ ABC, ∠ C = 90 °, BC: AC = 3:4, ab = 10, then AC =?
Because, ∠ C = 90 °,
According to Pythagorean theorem, BC ^ 2 + AC ^ 2 = AB ^ 2 = 100,
Because BC: AC = 3:4,
So BC = 3aC / 4,
So (3aC / 4) ^ 2 + AC ^ 2 = 100,
That is (25ac ^ 2) / 16 = 100,
So AC ^ 2 = 64, AC = 8
If the intersection of the image of the first order function y = K + X-1 and X-1 is on the positive half axis of the x-axis, then the value range of K is______ .
y=x+k-1，
X = 0, k = 1,
The coordinates of the intersection point between the image of the first-order function and the x-axis are (1-k, 0),
∵ the intersection point of the image of the linear function y = K + X-1 and the x-axis is on the positive half axis of the x-axis,
∴1-k＞0，
∴k＜1．
So the answer is k < 1
(2012 · Linyi) as shown in the figure, if the point m is any point on the positive half axis of the x-axis, the PQ ∥ Y-axis is made through the point m, and the functions y = k are respectively intersected (2012 · Linyi) as shown in the figure, if the point m is any point on the positive half axis of the X axis, the PQ ∥ Y axis is made through the point m, and the function y is intersected respectively= K1X (x > 0) and y= K2x The following conclusion is correct () A. ∠ poq cannot be equal to 90 ° b.pmqm= The graph of these two functions must be symmetric about the x-axis. The area of △ poq is 12 (| K1 | + | K2 |)
The correct answer is d
(2012 · Shandong) as shown in the figure, the geometry e-abcd is a pyramid, △ abd is a regular triangle, CB = CD, EC ⊥ BD (I) confirmation: be = de; (II) if ∠ BCD = 120 ° and M is the midpoint of line AE, it is proved that DM ‖ plane bec
(1) It is proved that: ∵ tetrapyramid e-abcd, base ⊥ abd is an equilateral triangle, CB = CD, △ BCD is isosceles triangle, take the midpoint o of BD, connect AC, O on AC ∵ EC ⊥ BD ≁ EO ⊥ bottom surface in O, AC ⊥ BD ᙨ bed is isosceles triangle
As shown in the figure, the image of the quadratic function y = ax2-4x + C passes through the coordinate origin and intersects with the X axis at point a (- 4,0) (1) Find the analytic formula of quadratic function; (2) If there is a point P on a parabola, s △ AOP = 8, please write out the coordinates of point P directly
(1) According to the known conditions, C = 0A × (- 4) 2-4 × (- 4) + C = 0, and a = - 1C = 0. Therefore, the analytic formula of the quadratic function is y = - x2-4x; (2) ∵ the coordinates of point a are (- 4, 0),
RELATED INFORMATIONS
- 1. As shown in the figure, the image of the quadratic function y = ax2-4x + C passes through the coordinate origin and intersects with the X axis at point a (- 4,0) (1) Find the analytic formula of quadratic function; (2) If there is a point P on a parabola, s △ AOP = 8, please write out the coordinates of point P directly
- 2. As shown in the figure, the chord AB and radius OC of ⊙ o are extended and intersected at point D, BD = OA. If ∠ AOC = 105 °, then ∠ D=______ Degree
- 3. As shown in the figure, ad is the height on the hypotenuse of the right angle △ ABC, de ⊥ DF, and de and DF intersect AB, AC and E, f respectively AD＝BE BD．
- 4. It is known that in △ ABC, ab = AC, BD and CE are the heights on the edge of AC and ab respectively, connecting De Results: (1) △ abd ≌ △ ace; (2) The quadrilateral BCDE is isosceles trapezoid
- 5. It is known that ad is the median line of the triangle ABC, e is the extension of CE at any point on ad, and the intersection of AB and F is proved to be AE / ad = 2AF / BF
- 6. As shown in the figure, the triangle ABC and the triangle ECD are isosceles right triangles, in which AC = BC, DC = EC, and C on ad, connect AE, de. please find out a pair of congruent triangles in the graph, and write the process of proving their congruence
- 7. As shown in the figure, in RT triangle ABC, angle c = 90 °, angle 1 = equal to angle 2, CD = 1.5, BD = 2.5?
- 8. In the RT triangle ABC, if the angle ACB = 90 degrees, BC = 3, AC = 4, the vertical bisector De of AB intersects the extension of BC at point E, then
- 9. As shown in the figure, in △ ABC, ∠ ACB = 90 °, points D and E are on AB, ad = AC, be = BC. Try to judge whether the size of ∠ DCE is related to the degree of ∠ B. if so, ask for the relationship between them; if not, please determine the degree and explain the reason
- 10. As shown in the figure, in △ ABC, ab = AC, ad ⊥ BC at D, ﹤ bad = 40 °, point E on AC, and AE = ad, calculate the degree of ∠ CDE ditto
- 11. As shown in the figure, in △ ABC, ∠ BAC = 90 °, ad ⊥ BC at D, e is a point on AB, AF ⊥ CE is at F, ad intersects CE at G point. It is proved that ∠ B = ∠ CFD
- 12. As shown in the figure, in RT △ ABC, ∠ C = 90 °, BC = AC, points D and E are on BC and AC respectively, and BD = CE, M is the midpoint of AB, and △ MDE is isosceles triangle As shown in the figure, in RT △ ABC, ∠ C = 90 °, BC = AC, points D and E are respectively on BC and AC, and BD = CE, M is the midpoint of AB, and △ MDE is isosceles right triangle. Please explain the reasons
- 13. As shown in the figure, in the triangle ABC, BC = 6, e and F are the midpoint of AB and AC respectively, the moving point P is on the ray EF, BP intersects CE at D, bisector of angle CBP intersects CE at Q, when CQ = 1 / 3 * ce, EP + BP=____ .
- 14. As shown in the figure, in the right angle trapezoid ABCD, ad ‖ BC, ∠ ABC = 90 °, BD ⊥ DC, BD = DC, CE bisection ∠ BCD, crossing AB at point E, crossing BD at point h, en ∥ DC intersecting BD at point n ①BH=DH；②CH=( 2+1)EH；③S△ENH S△EBH＝EH EC. The correct one is () A. ①②③ B. Only ② and ③ C. Only two D. Only 3
- 15. As shown in the figure, in the isosceles △ ABC, ab = AC. the semicircle with diameter AB intersects BC at point D and AC at point E De = 40 °, find ∠ A and Degree of AE
- 16. Let d be a point on the edge BC of △ ABC, and AB2 = ad2 + BD × DC
- 17. It is known that in the triangle ABC, D is a point on AB, be is perpendicular to ad, CF is perpendicular to ad, and perpendicular feet are e and f respectively If ad is the central line of the triangle ABC, be = CF is proved If ABC is a triangle of ABC, prove that it is a triangle fast
- 18. As shown in the figure, in △ ABC, it is known that ab = BC = Ca, AE = CD, ad and be intersect at point P, BQ ⊥ ad at point Q. verification: BP = 2pq
- 19. It is known that, as shown in the figure, in △ ABC, ad and AE are the bisectors of the height and angle of △ ABC respectively. If ∠ B = 30 ° and ∠ C = 50 °, calculate the degree of ∠ DAE
- 20. As shown in the figure, it is known that ad is the bisector angle of the isosceles triangle ABC, and the angle c = 90 degrees proves that ab = AC + CD