Let's break down the problem step by step:

Step 1:

We are given a right triangle ABC at vertex A, with altitude AH and median AD. We also know that I and J are the points where the medians of triangles ABH and ACH intersect with each other.

Step 2:

Since triangle ABC is a right triangle, we know that angle A is a right angle (90°). Therefore, we can conclude that triangle ABE is also a right triangle (with angle ABE being a right angle).

Step 3:

Now, let's focus on triangle ABH. Since I is the point where the median of triangle ABH intersects with the line segment AB, we know that AI = IB (by definition of median). Similarly, since J is the point where the median of triangle ACH intersects with the line segment AC, we know that AJ = JC (by definition of median).

Step 4:

Using the fact that I and J are on opposite sides of angle ABE, we can write:

AI + IB = AJ + JC

Since AI = IB and AJ = JC, we can simplify this equation to:

2IB = 2JC

Step 5:

Now, let's look at the triangles ABE and ACE. Since they share side AE and angle E is common to both triangles, we can say that:

∠EAB = ∠ECA (common angles)

Using this fact, we can conclude that:

AE = EB (since opposite sides of equal angles are equal)

Step 6:

Now we have:

AE = EB and IB = JC

Using these two equations, we can write:

IJ = IB - JC = AE - AE = 0

So, IJ is a zero-length line segment!

Conclusion:

Since IJ is a zero-length line segment, it means that I and J coincide with each other. This implies that:

IJ ⊥ AD (I and J are collinear with AD)

Therefore, we have shown that triangle ABE is a right triangle and IJ is perpendicular to AD.

Answer:

a. Tam giác ABE vuông b) IJ vuông góc với AD

làm ơn giúp mình với

\(\dfrac{120^5}{40^5}+\dfrac{8^{13}}{4^{10}}-\dfrac{390^4}{130^4}\)

\(=3^5+\dfrac{2^{39}}{2^{20}}-3^4\)

\(=243+2^{19}-81=524450\)

a) Căc cặp góc so le trong là:

\(\widehat{R_1}\) và \(\widehat{S_3}\)

\(\widehat{R_2}\) và \(\widehat{S_4}\)

Các cặp góc đồng vị là: 

\(\widehat{S_1}\) và \(\widehat{R_1}\)

\(\widehat{S_2}\) và \(\widehat{R_2}\)

\(\widehat{S_3}\) và \(\widehat{R_3}\)

\(\widehat{S_4}\) và \(\widehat{R_4}\)

Các cặp góc trong cùng phía là:

\(\widehat{S_4}\) và \(\widehat{R_1}\)

\(\widehat{S_3}\) và \(\widehat{R_2}\)

b) Ta có: 

\(\widehat{R_2}=120^o=>\widehat{S_2}=120^o\) (đồng vị)

\(\widehat{R_2}=120^o=>\widehat{S_3}=180^o-\widehat{R_2}=180^o-120^o=60^o\) (trong cùng phía) 

\(\widehat{S_4}=120^o=>\widehat{R_4}=120^o\) (đồng vị) 

\(\widehat{S_4}=120^o=>\widehat{R_1}=180^o-\widehat{S_4}=180^o-120^o=60^o\) (trong cùng phía)  

\(\widehat{R_1}=60^o=>\widehat{S_1}=60^o\) (đồng vị)  

\(\widehat{S_3}=60^o=>\widehat{R_3}=60^o\) (đồng vị) 

a) Ta có:

\(\left\{{}\begin{matrix}AB\perp AC\\KH\perp AC\end{matrix}\right.=>AB//KH\) 

b) Ta có: 

\(\widehat{ABK}=\widehat{BKI}\left(=60^o\right)\)

Mà hai góc này ở vị trí so le trong 

=> AB//KI 

c) AB//HK = > \(\widehat{ABK}+\widehat{HKB}=180^o\)

Mà: \(\widehat{ABK}=\widehat{BKI}\) 

\(=>\widehat{BKI}+\widehat{HKB}=180^o\)

=> \(\widehat{HKI}\) là góc bẹt hay H, K, I thẳng hàng

a) Ta có:

\(\widehat{ADE}=\widehat{ABC}\left(=45^o\right)\)

Mà hai góc này ở vị trí đồng vị

=> DE//BC

b) Ta có: 

\(\widehat{FEC}=\widehat{ECB}\left(gt\right)\)

Mà hai góc này ở vị trí so le trong 

=> EF//BC

c) Ta có: DE//BC

=> \(\widehat{DEC}+\widehat{ECB}=180^o\) (trong cùng phía) 

Mà: \(\widehat{FEC}=\widehat{ECB}\left(gt\right)\)

\(=>\widehat{FEC}+\widehat{ECB}=180^o\)

\(=>\widehat{DEF}\) là góc bẹt

=> D, E, F thẳng hàng

a) Ta có: 

\(\widehat{MAB}=\widehat{ABC}\left(=55^o\right)\)

Mà hai góc này ở vị trí so le trong 

=> AM//BC 

b) Ta có:
\(\widehat{NAC}=\widehat{ACB}\left(=40^o\right)\)

Mà hai góc này ở vị trí so le trong

=> AN//BC 

c) Xét tam giác ABC có:

\(\widehat{BAC}+\widehat{ABC}+\widehat{ACB}=180^o\\ =>\widehat{BAC}=180^o-\widehat{ABC}-\widehat{ACB}\\ =>\widehat{BAC}=180^o-55^o-40^o=85^o\) 

\(\widehat{MAB}+\widehat{BAC}+\widehat{NAC}=55^o+85^o+40^o=180^o\) 

=> \(\widehat{MAN}\) là góc bẹt => M, A, N thẳng hàng

\(\left(8+2\dfrac{1}{3}-\dfrac{3}{5}\right):\left(5-\dfrac{1}{4}-\dfrac{5}{8}\right)\\ =\left(8+2+\dfrac{1}{3}-\dfrac{3}{5}\right):\left(5-\dfrac{2}{8}-\dfrac{5}{8}\right)\\ =\left(10+\dfrac{1}{3}-\dfrac{3}{5}\right):\left(5-\dfrac{7}{8}\right)\\ =\left(\dfrac{150}{15}+\dfrac{5}{15}-\dfrac{9}{15}\right):\left(\dfrac{40}{8}-\dfrac{7}{8}\right)\\ =\dfrac{146}{15}:\dfrac{33}{8}\\ =\dfrac{146}{15}\cdot\dfrac{8}{33}\\ =\dfrac{1168}{495}\)

\(f\left(x\right)+g\left(x\right)=x^2h\left(x\right)\)

\(2x^3-3x^2+3x+1+g\left(x\right)=x^2\left(2x+1\right)\)

\(g\left(x\right)=2x^3+x^2-2x^3+3x^2-3x-1=4x^2-3x-1\)

chọn C

`#3107.101107`

Ta có:

`f(x) + g(x) = x^2h(x)`

`\Rightarrow g(x) = x^2h(x) - f(x)`

`g(x) = x^2 * (2x + 1) - (2x^3 - 3x^2 + 3x + 1)`

`= 2x^3 + x^2 - 2x^3 + 3x^2 - 3x - 1`

`= 4x^2 - 3x - 1`

Chọn C.