Cho tam giác ABC, đường cao AH, trung tuyến AM. Điểm D đối xứng với A qua H, E đối xứng với A qua M.
a) C/m: tứ giác ABEC là hình bình hành.
b) C/m: tứ giác BCED là hình thang cân.
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5. (2 + x) (x+2) - \(\dfrac{1}{2}\) . (6-8x)2 + 17
= 5 ( x2 + 4x + 4) - \(\dfrac{1}{2}\). 4 ( 3 - 4x) 2+ 17
= 5x2 + 20 x + 20 - 2 ( 9 - 24x + 16x2) + 17
= 5x2 + 20x + 20 - 18 + 58x - 32x2 + 17
= -27x2 + 78 x + 19
A = x2 + 5x - 5
A = x2 + 2. \(\dfrac{5}{2}\) x + ( \(\dfrac{5}{2}\))2 - \(\dfrac{45}{4}\)
A = (x + \(\dfrac{5}{2}\))2 - \(\dfrac{45}{4}\)
(x + \(\dfrac{5}{2}\))2 ≥ 0 ⇔ A = (x + \(\dfrac{5}{2}\))2 - \(\dfrac{45}{4}\) ≥ \(\dfrac{-45}{4}\)
A(min) = -45/4 = xảy ra ⇔ x + 5/2 = 0 ⇔ x = -5/2
I'll let you draw the figure.
a) AM is a median of the triangle ABC. Therefore, M is the midpoint of BC.
E is symmetric to A through M, so M is the midpoint of AE.
Consider the quadrilateral ABEC, it has M is the midpoint of both AE and BC. Thus, ABEC is a parallelogram.
b) Consider the triangle ADE, M is the midpoint of AE, H is the midpoint of AD. Therefore, HM is the average line of this triangle. This means \(HM//DE\) or \(DE//BC\), which means BCED is a trapezoid.
Triangle ABD has the height BH, which is also a median. Thus, ABD must be an isosceles triangle, which means the height BH is also a bisector, or \(\widehat{ABH}=\widehat{DBH}\) or \(\widehat{ABC}=\widehat{DBC}\)
On the other hand, ABEC is a parallelogram. So, \(AB//CE\) and this leads to \(\widehat{ABC}=\widehat{ECB}\) (2 staggered angles of 2 parallel lines)
From these, we have \(\widehat{DBC}=\widehat{ECB}\left(=\widehat{ABC}\right)\)
The trapezoid BCED (DE//BC) has \(\widehat{DBC}=\widehat{ECB}\). Therefore, BCED must be an isosceles trapezoid.