đường thằng (d) tiếp xúc với (O) tại A => D là tiếp tuyến của A

=> AM _|_ AB (tính chất tiếp tuyến) => tam giác AMB vuông A

lại có góc ANB=90^o (góc nội tiếp chắn nửa đường tròn) => tam giác ANB vuông tại N

xét tam giác vuông AMB và ANB có \(\widehat{B}\)chung

=> tam giác AMB đồng dạng với tam giác ANB => \(\frac{AB}{BM}=\frac{BN}{AB}\Rightarrow AB^2=BN\cdot BM\)

mà AB=2R không đổi => AB²=4R² không đổi => BM.BN=4R² không đổi

b) ta có \(\widehat{AQP}=\frac{1}{2}\left(sđAB-sđAP\right)=\frac{1}{2}sđPB\)(định lý góc côc định ngoài đường tròn)

lại có \(PNB=\frac{1}{2}sđPB\)(tính chất góc nội tiếp) => \(AQP=PNB\left(=\frac{1}{2}sđPB\right)\)

hay \(\widehat{MQP}=\widehat{PNB}\)mà \(\widehat{MNP}+\widehat{PNB}=180^o\)(kề bù) => ^MQP=^MNP=180⁰

=> tứ giác MNPQ nội tiếp 

c) áp dụng bđt Cosi cho 2 số dương ta có:

\(BM+BN\ge2\sqrt{BM\cdot BN}=2\sqrt{4R^2}=4R\)

dấu "=" xảy ra khi BM=BN <=> M trùng với N trái với giả thiết => BM+BN >4R(1)

chứng minh tương tự ta có BP+BQ >4R (2)

từ (1) và (2) => BM+BN+BP+BQ >8R (đpcm)

tôi cũng nghĩ là dùng Phương pháp dồn biến tìm MAX , MIN

\(P=\left(4x^2+3y\right)\left(4y^2+3x\right)+25xy\)

\(=16x^2y^2+12\left(x+y\right)\left(x^2-xy+y^2\right)+34xy\)

\(=16x^2y^2+12\left[\left(x+y\right)^2-2xy\right]+22xy\)

\(=16x^2y^2-2xy+12\)

Đặt  \(t=xy\Rightarrow B=16t^2-2t+12=16\left(t-\frac{1}{16}\right)^2+\frac{191}{16}\ge\frac{191}{16}\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}x+y=1\\xy=\frac{1}{16}\end{cases}\Leftrightarrow}\hept{\begin{cases}x=\frac{2+\sqrt{3}}{4}\\y=\frac{2-\sqrt{3}}{4}\end{cases}}\) hoặc  \(\hept{\begin{cases}x=\frac{2-\sqrt{3}}{4}\\y=\frac{2+\sqrt{3}}{4}\end{cases}}\)

Vậy  \(B_{min}=\frac{191}{16}\Leftrightarrow\left(x;y\right)=\left(\frac{2+\sqrt{3}}{4};\frac{2-\sqrt{3}}{4}\right);\left(\frac{2-\sqrt{3}}{4};\frac{2+\sqrt{3}}{4}\right)\)

3.(10.x)=111

=> 10x = 111 : 3

=> 10 x = 37

=> x = 37 : 10

=> x = \(\frac{37}{10}\)

3.(10.x)=111

    (10.x)=111:3

    (10.x)=37

          x=37:10

          x=3,7

vậy x=3,7

ta có:\(\widehat{aOb}\) = 180

\(\Rightarrow\)3 x \(\widehat{aOc}\)=180

\(\Rightarrow\)\(\widehat{aOc}\)=180 : 3 = 60

\(\Rightarrow\)\(\widehat{aOc}\)=\(\widehat{bOd}\)= 60 (2 góc đối đỉnh)

ta có: \(\widehat{aOc}\)+\(\widehat{cOb}\)= 180 (2 góc kề bù)

\(\Rightarrow\)60 + \(\widehat{cOb}\)= 180

\(\Rightarrow\)\(\widehat{cOb}\)= 180 - 60 = 120

\(\Rightarrow\)\(\widehat{aOd}\)=\(cOb\)= 120 (2 goc đối đỉnh)

Vậy \(\widehat{aOc}\)= 60;\(\widehat{cOb}\)= 120;\(\widehat{bOd}\)= 60;\(\widehat{aOd}\)=120

cảm ơn bạn

1)

Ta có : \(\frac{a}{3}=\frac{b}{4}=\frac{c}{5}\)=> \(\frac{a^2}{9}=\frac{b^2}{16}=\frac{c^2}{25}\)=> \(\frac{a^2}{9}=\frac{2b^2}{32}=\frac{c^2}{25}\)

Đặt \(\frac{a^2}{9}=\frac{2b^2}{32}=\frac{c^2}{25}=k\)

=> \(\hept{\begin{cases}a^2=9k\\2b^2=32k\\c^2=25k\end{cases}}\)

=> \(a^2+2b^2-c^2=9k+32k-25k=16k\)

=> \(16k=144\)

=> \(k=9\)

Do đó \(\hept{\begin{cases}a^2=9\cdot9\\2b^2=32\cdot9\\c^2=25\cdot9\end{cases}}\Rightarrow\hept{\begin{cases}a^2=81\\b^2=144\\c^2=225\end{cases}}\Rightarrow\hept{\begin{cases}a=9\\b=12\\c=15\end{cases}}\)

2) Ta có : \(\frac{a}{5}=\frac{b}{7}=\frac{c}{9}\)=> \(\frac{a^2}{25}=\frac{b^2}{49}=\frac{c^2}{81}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có :

\(\frac{a^2}{25}=\frac{b^2}{49}=\frac{c^2}{81}=\frac{a^2+b^2-c^2}{25+49-81}=\frac{-28}{-7}=4\)

=> \(\hept{\begin{cases}\frac{a^2}{25}=4\\\frac{b^2}{49}=4\\\frac{c^2}{81}=4\end{cases}}\Rightarrow\hept{\begin{cases}a^2=100\\b^2=196\\c^2=324\end{cases}}\Rightarrow\hept{\begin{cases}a=10\\b=14\\c=18\end{cases}}\)

a) đặt \(\frac{a}{3}=\frac{b}{4}=\frac{c}{5}=k\Rightarrow\hept{\begin{cases}a=3k\\b=4k\\c=5k\end{cases}}\)

đặt \(a^2+2b^2-c^2=144\)

\(\Leftrightarrow\left(3k\right)^2+2\left(4k\right)^2-\left(5k\right)^2=144\)

\(\Leftrightarrow9k^2+32k^2-25k^2=144\)

\(\Leftrightarrow k^2\left(9+32-25\right)=144\)

\(\Leftrightarrow k^216=144\)

\(\Leftrightarrow k^2=9\)

\(\Leftrightarrow k=\sqrt{9}=\pm3\)

do đó 

\(\frac{a}{3}=k\Leftrightarrow\frac{a}{3}=\pm3\Rightarrow\hept{\begin{cases}a=3.3=9\\a=3.\left(-3\right)=-9\end{cases}}\)

\(\frac{b}{4}=k\Leftrightarrow\frac{b}{4}=\pm3\Rightarrow\hept{\begin{cases}b=4.3=12\\b=4.\left(-3\right)=-12\end{cases}}\)

\(\frac{c}{5}=k\Leftrightarrow\frac{c}{5}=\pm3\Rightarrow\hept{\begin{cases}c=5.3=15\\c=5.\left(-3\right)=-15\end{cases}}\)

vậy các cặp a,b,c thỏa mãn là \(\left\{a=9;b=12;c=15\right\}\left\{a=-9;b=-12;c=-15\right\}\)

1 Columbus ..discovered................(discover)America more than 400 years ago.

2 They ....came..........(come) here a month ago. Albert Einstein ....died.............(die) on 18,1995

3 We .went........(go) to the mountains last Sunday.

4 They .....came........(come) here a month ago.

6 Nam.....bought............(buy) several records last week.

7 .........she.............(do) her homework last night?

8 My father ....built...............(build) this house five years ago.

9 The factory......began...........(begin) operationlast year.

10 What....did.....you...do.......(do) yesterday?

11 They.....finished............(finish) their dinner half an hour ago.

12 We (study)......studied..........a very hard lesson the day before yesterday.

13 I (read).......have read..............that novel by Hemingway several times before.

14 We (study)...have studied......almost every lesson in this book so far.

15 My wife and I (travel)...travelled...........to Mexico by air last summer.

16 I (have)...had............a little trouble with my car last week.

17 What did you (do)...do...........yesterday?

18 How long have you (learn)........learnt..........English?

19 Tom (never be)....has never been.............in Hanoi.

20 The plane (stop)...has.stopped...........at a small town.

21 It (take)...has.taken..............off immediately after refueling...............

22 She (be).was.........so happy when she (hear)....heard........the news that she (cry)..cried............

23 After class, one of the students always (erase)....erases.....the chalk board.

1 We (study).....studied...........a very had lesson the day before yesterday.

2 I (read)....have read..........that novel by Hemingway several times before.

3 We (study) ....has studied................almost every lesson in this book so far.

4 My wife and I (travel)........traveled.............to Mexico by air last summer.

5 I (have)....have...............a little trouble with my car last week.

6 What did you (do).....do........yesterday ?

7 How long have you (learn)..........learnt............English ?

8 Tom never (be).....has never been...........in Hanoi.

9 The plane (stop)...stops..........at a small town. It then (take)......................off immediately after refuelling.

10 She (be)....was................so happy when she (hear).....heard........,...the news that she (cry)..cried..............

11 Steven (live)......has lived..........in London since 1990.

12 We (leave)...left.............Singapore six months ago.

13 I never (eat)...have never eaten...........sneak meat.

14 Jack (meet)...met...........Simon for dinner last night

15 Oh no ! Someone (steal).....has stolen.......my bag.

1 We studied a very had lesson the day before yesterday.

2 I have read that novel by Hemingway several times before.

3 We have studied almost every lesson in this book so far.

4 My wife and I traveled to Mexico by air last summer.

5 I had a little trouble with my car last week.

6 What did you do yesterday ?

7 How long have you learned English ?

8 Tom has never been in Hanoi.

9 The plane stopped at a small town. It then took off immediately after refuelling.

10 She was so happy when she heard the news that she cried

11 Steven has lived in London since 1990.

12 We left Singapore six months ago.

13 I have never eaten sneak meat.

14 Jack met Simon for dinner last night

15 Oh no ! Someone stolen my bag.

1. have known

2. have lived

3.has written

4.hasn't broken

5.has broken

6. have seen

7.has had

8.have been

9. haven't finished

10. have been

Cậu có thể tham khảo bài làm trên đây ạ, mn ai thấy đúng thì cho mk xin 1 t.i.c.k ạ, thank nhièu

                                                             # HỌC TỐT NHA #

1 Have known

2 have lived

3 has written

4 haven't broken

5 broken

6 have seen

7 had

8have been

9 haven't finished

10 have been

1. for
2. since
3. .....
4. since
5. sĩne
6. since
7. for
8. since
9. for​
10. since
11. since
12. since
13. for
14. since
15. since
16. for
17. since
18. since
19. for
20. since
21. since
22. for
23. for
24. for
25. since

Cậu có thể tham khảo bài trên đây ạ, nếu ai thấy đúng cho mk xin 1 t.i.c.k, thank nhiều

Bổ sung và sửa lại cho bạn trước:

3. for (Cụm từ for ages = long time)

15. for (eight years là một khoảng thời gian)

Ta có : 4³⁰ = (2²)³⁰ = 2^2.30 = 2⁶⁰

Lại có : 8²⁰ = (2³)²⁰ = 2^3.20 = 2⁶⁰

Vì 2⁶⁰ = 2⁶⁰

=> 4³⁰ = 8²⁰

4³⁰ = ( 4³ )¹⁰ = 64¹⁰

8²⁰ = ( 8² )¹⁰ = 64¹⁰

=> 4³⁰ = 8²⁰