làm đi '-'
I. Find the word which has a different sound in the part underlined.
1. A. cleanup B. teach C. beach D. create
2. A. trash B. bag C. jar D. plastic
3. A. horror B. boring C. order D. movie
4. A. hiking B. climbing C. campsite D. rafting
5. A. flower B. towel C. pillow D. shower
6. A. buildings B. rivers C. stores D. parks
7. A. bakery B. station C. arcade D. café
8. A. spacesuit B. gravity C. astronaut D. bag
9. A. general B. elephant C. went D. become
10. A. animated B. fighting C. fiction D. ticket
11. A. kayak B. camera C. battery D. campsite
12. A. mind B. different C. win D. invader
13. A. country B. underground C. sound D. south
14. A. wildlife B. office C. sticker D. river
15. A. reuse B. purpose C. community D. computer
16. A. kayak B. country C. friendly D. canyon
17. A. exercise B. lock C. space D. device
18. A. market B. arcade C. park D. library
19. A. thinks B. earthscrapers C. megacities D. homes
20. A. canyon B. kayak C. cave D. campsite
21. A. flashlight B. thirsty C. hiking D. night
22. A. cathedral B. weather C. there D. although
23. A. attack B. battle C. animated D. army
24. A. think B. theater C. clothes D. thank
25. A. clothes B. throw C. overseas D. bottle
26. A. charity B. donate C. trash D. animal
27. A. drama B. action C. animated D. fantastic
28. A. police B. hospital C. office D. opposite
29. A. gravity B. float C. future D. station
30. A. opera B. palace C. machine D. statue
II. Pick out the word whose main stressed syllable is different from the rest in the same line.
1. A. spacesuit B. shower C. however D. special
2. A. underground B. technology C. underwater D. eco-friendly
3. A. environment B. community C. cleanup D. recycle
4. A. paper B. plastic C. bottle D. protect
5. A. comedy B. enjoy C. drama D. theater
6. A. battery B. amazing C. blanket D. flashlight
7. A. vacation B. valley C. rafting D. mountain
8. A. against B. between C. soldier D. attack
9. A. police B. market C. office D. theater
10. A. charity B. ocean C. protect D. donate
11. A. newspapers B. community C. recycle D. reuse
12. A. action B. horror C. suggest D. science
13. A. soldier B. invader C. battle D. army
14. A. museum B. cathedral C. vacation D. opera
15. A. swimming B. inside C. trekking D. campsite
16. A. bottle B. jacket C. pillow D. remember
17. A. earthscraper B. megacity C. eco-friendly D. future
18. A. opposite B. between C. subway D. station
19. A. palace B. hotel C. mountain D. statue
20. A. astronaut B. gravity C. station D. explore
III. Choose the best option to complete each of the following sentences.
1. My dream house will be ………..the sea, so in my room I can see beautiful fish and other sea animals in front of, behind, above or below me. That will be amazing!
A. by B. on C. near D. under
2. A: Should we bring some snacks to eat? B: ………..
A. I don’t have. B. Yes, you need. C. Yes, good idea. D. Yes, I did.
3. You should recycle empty cans and bottles. Don’t just ………..
A. reuse them B. donate it C. throw it D. throw them away
4. I think the British Museum is ………..museum in London.
A. the most popular B. popular C. more popular D. the more popular
5. You ………..visit the museum today. It’s closed at the moment.
A. need B. needn’t C. can’t D. shouldn’t
6………..in this lake. It’s very deep and dangerous.
A. Don’t swim B. Swimming C. You can swim D. Swim
7. There’s ………..new girl in our class. ………..girl’s brother is in Dan’s class.
A. the/ A B. a/ The C. the/ The D. a/ A
8. My birthday is ………..the summer, ………..8th July.
A. in/ at B. on/ on C. in/on D. at/ in
9. Now we drive petrol cars, but in 2040, all transport ………..electric.
A. was B. have to be C. will be D. is
10. People ………..on Mars someday, but it is too expensive to travel there now.
A. are living B. will live C. lived D. live
11. I ………..Camden, the most famous street market in London if I ………..time.
A. will visit/ should have B. visit/ will have C. will visit/ have D. visit/ have
12. I think technology ………..make the world a better place.
A. is going to B. have to C. won’t D. will
13. A: I believe everyone can help save the environment. B: ………..
A. What’s wrong? B. Yes, I’d love to. C. Yes, I think. D. I agree.
14. I don’t like going ………..I’m scared of traveling down a river in a thin boat.
A. swimming B. hiking C. trekking D. kayaking
15. My friends, Susie and Mike, ………..in London last weekend.
A. is B. was C. were D. are
16. A: How might homes change in the future? B: ………..
A. I think life is different now from in the past.
B. We might use our mobiles to turn on everything in our homes.
C. Scientist would find life on other planets.
D. Many people lived under the sea.
17. A: What is the largest country in the world? B: ………..
A. There are many large countries in the world. B. Russia is a large country.
C. Russia is larger than China. D. I think Russia is the largest.
18. The Dragon ………..is one of the most popular tourist attractions in Da Nang. It is 666m long and crosses the Han river.
A. bridge B. statue C. park D. tower
19. Ratatouille is a/an ………..movie. The characters are drawings, not real people.
A. science fiction B. action C. animated D. drama
20. In 20 years from now, I think scientists might be able to print new human body parts using ………..
A. 3D printers B. smart homes C. automatic food machine D. drones
21. In 2030, ………..will deliver most things in less than half an hour.
A. drones B. 3D printers C. smart walls D. smart screens
22. You should wear ………..to protect your eyes from the bright sun.
A. towels B. tents C. jackets D. sunglasses
23. A ………..will help you see in the dark.
A. flashlight B. drama C. battery D. horror
24. I like going to the ………..because I can read and borrow many interesting books there.
A. department store B. school C. bookstore D. library
25. In 10 years’ time, I think people will wear ………..clothes. These clothes will change color or shape when it’s too hot or cold.
A. technology B. smart C. eco-friendly D. friendly
26. Nowadays, we can use some computers by touching their ………..
A. smart walls B. printers C. screens D. helpers
27. You can ………..old books and computer equipment to charity.
A. reuse B. get C. protect D. donate
28. I need to go to the ………..to send these letters to my grandparents.
A. police station B. post office C. supermarket D. library
29. The ………..of King Leonidas of Sparta in Greece is bigger and taller than a real man.
A. museum B. tower C. statue D. bridge
30. Speaker 1: Excuse me. Can I have the check, please? Speaker 2: ………..
A. Yes, you can. B. Yes, here you are. C. Yes, you would. D. Yes, please.
31. You should visit Pongour ………..in Lâm Đồng. It’s wonderful to see water fall from a high place.
A. cave B. island C. beach D. waterfall
32. I think people in Brazil are ………..people in the UK.
A. the most friendly B. the friendliest C. more friendly D. friendlier than
33. Disneyland is one of the biggest ………..in the USA. It’s full of magic, roller coasters and entertainment for all ages.
A. museums B. buildings C. shopping malls D. amusement parks
34. William can’t watch ………..movies. He thinks they’re too scary and he always has bad dreams after watching them.
A. drama B. action C. comedy D. horror
35. Mont Blanc is a very high ………..in Europe. It’s 4,809 meters above sea level
A. canyon B. forest C. mountain D. highland
36. The weather here is …….. than in my hometown. It’s too cold here.
A. better B. best C. worst D. worse
37. We have to ………..wild animals, such as whales or tigers, or they will disappear.
A. look B. take part in C. pick up D. protect
38. The Leaning Tower of Pisa is one of Italy’s most famous ………..
A. building B. attraction C. malls D. landmarks
39. There is a ………..near my school. Homeless people go there for free meals and a place to stay.
A. police station B. hospital C. organization D. charity
40. At the concert, you can get food and drink for ………..You don’t have to pay money for it.
A. charity B. reuse C. free D. recycle
41. Napoleon Bonaparte was a great leader and he won many ………..
A. battles B. leaders C. soldier D. army
42. There is no ……….., so everything floats in space.
A. space station B. devices C. gravity D. space suit
43. Trưng Trắc and Trưng Nhị were Vietnamese great female ………..
A. armies B. soldier C. generals D. invaders
44. You need to bring a ………..so you don’t get cold at night.
A. pillows B. soap C. sleeping bag D. battery
45. I love sunbathing on sandy ………..
A. mountains B. islands C. beaches D. deserts
IV. Put the verbs in brackets into the correct form.
1. My brother (listen) ………………. to music now.
2. The people in the café (not be) ………………….. friendly when I was there yesterday.
3. – How …………………. homes (change) …………………. in the future?
– There (be) ………………………. smart devices inside it.
4. Minh (not go) …………..………jogging yesterday, she (stay) …………….……at home and (do) ………………..….aerobics.
5. If she (study) ………………..…. hard, she (get) ………………..…. good grades
6. The police station (be) ………………… opposite the bank.
7. Long is thirsty. He’d like (drink) …………….……….. some water.
8. It (be) ………………….. a great film in 2001.
9. ……………you (watch) ……………..………. the tennis match on TV last night?
10. If she (do) …………………. morning exercises, she (be ) …………..…. healthy.
11. I (be) ………………… tired. Let's (take) ………………… a taxi to get there.
12. They (visit) ……………………… Da Lat this summer vacation.
13. I (leave) ………………….. my school bag at school this morning.
14. Owen is a football coach, but he (not work) ……………….……..at the moment.
15. The train (arrive) ………………… in ten minutes.
16. If he (not / come) …………..…. to the party on time, everybody (leave) ……..…. there.
17. Mrs Thu (do) …………… the housework every day.
18. I (not be) ………………….. very happy yesterday.
19. When my brother and I (be) …………….…small, we (go) …………..…… swimming at the weekend.
20. How about (have) ………………… dinner in a restaurant?
21. He (not go) …………………………….fishing in the winter.
22. I went to the shop but I (not have) ………………….. any money.
23. My father (take) ………………..………me to the zoo 5 days ago.
24. What about ( go) …………………….….to Nha Trang?
25. Now they (play) ………………………soccer.
26. Our teacher (tell) ………………….. us to be quiet yesterday.
27. Quang (not play) …………..……….any sports. He (like) …………….….collecting stamps.
28. She (get) ……………..…. wet if she (not / bring ) ……………..…. a rain coat.
29. Lan (have) ……………… breakfast and (go) ……………to school at 6.30 a.m.
30. …………you usually ( jog) …….……in the morning? – Yes, I do.
31. Susan (not know) ………………….. about the exam and she did very badly.
32. …………….you (want) ……………………to go somewhere, Mr Long?
33. We (play) ………………… sports together at the sports center this Sunday.
34. If he (go) …………..…. on smoking, he (damage) ………………..…. his health.
35. This summer vacation, my parents (visit) …………………………Hue.
36. I (buy) ………………….. a ticket for the football match yesterday.
37. In the future, people (not play) …………………….…… individual games.
38. Vinh usually (go)………………..…..to school on foot.
39. …………your students (play) ………………soccer every afternoon?
40. I think a robot helper (do) ………………………. the washing and cleaning at home.
V. Write the correct forms of the words in the brackets.
1. You shouldn’t go hiking today. It’s raining hard, so it’s very ……………………… (DANGER)
2. You should visit Bản Giốc; it’s an ……………………… waterfall in the north of Vietnam. (AMAZE)
3. My little sister loves watching ……………………… movies. Her favorite ones are Toy Story series. (ANIMATION)
4. In some countries, people will live ……………………… in earthscrapers. (GROUND)
5. Visitors can enjoy the ……………………… beauty of Mộc Châu Highland. (NATURE)
6. It’s……………………… to live in big cities than in the countryside. (NOISE)
7. Trees are very ……………………… They give food to people and animals. (USE)
8. Act Green Vietnam is an ……………………… charity in Ho Chi Minh city. (ENVIRONMENT)
9. In the future, many people will live under the sea; so ……………………… cities will become popular. (WATER)
10. Living in a town is ……………………… from living in a big city. (DIFFER)
11. In the next 20 years, many homes might be smart homes with robot ……………………… to do the chores. (HELP)
12. It is a very ……………………… movie with lots of laughter and the ending is hilarious. (FUN)
13. Beijing, the capital of China, is an interesting city; it has the most popular tourist attractions and the ……………………… restaurants and cafés in the country. (GOOD)
14. When working outside in space, astronauts have to wear ……….…… (SPACE)
15. We can ……………………… empty glass jars to store food, like jam and nuts. (USE)
16. Last year, we went ……………………… on the Colorado River; it was wonderful. (RAFT)
17. A smart………………….machine will wash dirty clothes and make them ready for you to put on. (WASH)
18. Yesterday I watched a ……………………… movie about Napoleon Bonaparte on TV. (HISTORY)
19. We are taking these old newspaper to the ……………………… bins in town. (RECYCLE)
20. Many people will live in ……………………….in the future. (EARTH)
21. A(n) ……………………… house is a house that is good for the environment. (FRIEND)
22. China is the most ……………………… country in the world with more than 1,4 billion people. (POPULATION)
23. The downtown area of Shanghai is always ……………………… with hundreds of people, visitors, cars, buses and taxis. (CROWD)
24. Thank you for ……………………… me to your birthday party. (INVITE)
25. According to World Air Quality, China is more ……………………… than Vietnam. (POLLUTION)
26. You can help by ……………………… money to the charity. (DONATE)
27. In the future, many people will live in ……………………… - cities with a population of over ten million people. (CITY)
28. Quang Trung fought against ……………………… from the north and won the battles. (INVADE)
29. You should bring ……………………… water so you don’t get thirsty. (BOTTLE)
30. When we go camping, we have to sleep in …………………….bags. (SLEEP)
VI. Read the following passage and choose the best answer for each blank.
FAMOUS PEOPLE IN YOUR COUNTRY
1. Yen - Vietnam
Yesterday, I watched a documentary about one of our country's great leaders - General Vo Nguyen Giap. People all over the world (1) …………..him as General Giap. He was born in 1911 in Quang Binh Province and was a general in Vietnamese army. He led the army and won many (2)………, such as Lang Son (1950), Hoa Binh (1951-1952), and Dien Bien Phu (1954)
2. Anthony - England
In my country, we are proud (3) ……….Sir Winston Churchill. He was the Prime Minister during the World War II. Surprisingly, he was a great writer, and (4) …………the Nobel Prize for Literature. He was a painter too, and he created (5) ……. than 550 paintings.
1. A. knew B. know C. knows D. doesn't know
2. A. soldiers B. battles C. generals D. armies
3. A. in B. on C. of D. at
4. A. did B. put C. gave D. won
5. A. more B. as C. no D. so
VII. Circle the best answer A, B, C or D to complete the passage.
The hiking club went (1) ………a trip to Bear Mountain last Saturday. I signed up for it. The bus dropped us off at the bottom of the (2) ………., and we hiked until we got to a cabin halfway up the mountain. It took us seven hours (3)……..to the cabin, so it was late by the time we arrived. After we unpacked, we started a (4) …………. Then we sang, ate, and told stories around the fire. The following day we left at dawn and hiked to the top of the mountain. There was a lake on the mountain, and we went swimming in the lake. The water was cool and the scenery was great. After we (5) ………..pictures, we hiked back down. We had an enjoyable trip.
1. A. on B. to C. at D. by
2. A. river B. lake C. mountain D. hill
3. A. hiking B. to hike C. hiked D. hike
4. A. camping B. camp C. campsite D. campfire
5. A. taking B. to take C. took D. take
VIII. Read and decide T (true)/ F (false) for each statement.
The Battle of Bach Dang is an animated movie about Ngo Quyen, a great leader in Vietnamese history. In 938, a strong army from the Han Dynasty invaded Vietnam. Ngo Quyen led the Vietnamese forces and won the battle of Bach Dang River. Ngo Quyen told his soldiers to put sharp stakes into the bottom of the river. As the Han army sailed up Bach Dang River on large boats, the river's water level started to fall. The stakes in the river appeared and broke their boats. Ngo Quyen's army attacked them and easily won the battle. A year later, Ngo Quyen became the king and moved the capital city to Co Loa (today's Dong Anh, Ha Noi). The Battle of Bach Dang is a very interesting movie about Vietnamese history, and the animation is great.
1. The Battle of Bach Dang happened in 983. ………….
2. Ngo Quyen's soldiers put stakes in Bach Dang River. ………….
3. The Han army went to Vietnam on foot. ………….
4. Ngo Quyen became a king in 939. ………….
5. Ngo Quyen moved the capital city to Co Loa. ………….
IX. Read the passage below and choose the correct answer for each question.
Most people think that the Middle East is only a desert. This is far from true. Last year, I went hiking in Iran. Iran has many mountains. You can find trees that lose their leaves in winter. If you’re lucky, you may see a yellow deer or a brown bear, two species that live there. The air is fresh, and you can swim in clear, clean lakes. There are so many birds to see. It is fun to go into the forest with binoculars to look at birds. Seeing the birds through binoculars makes them seem much nearer than they actually are. You can write down all of the species that you see. You should learn about their calls first. That means the sounds that each bird makes. Sometimes it’s hard to see the birds. However, you can identify them if you happen to hear them. I want to go back to Iran again someday. It’s so beautiful that I want to see and explore much more of it!
1. It's not true that the Middle East is only …………
A. a dessert B. a desert C. a mountain D. a country
2. Two kinds of animals that live in Iran are……….
A. birds and brown bears B. birds and yellow deer
C. yellow deer and brown bears D. deer and bears
3. Going into the forest with binoculars to look at birds is ……….
A. awful B. terrible C. boring D. interesting
4. The word identify has the same meaning as……
A. recognize B. see C. look D. watch
5. The passage is about……..
A. mountains in Iran B. forests in Iran C. birds in Iran D. a trip to Iran
X. Read the passage then choose the correct answers.
In the future, we will live in amazing new homes. Our houses will use the sun or the wind to make electricity. We won’t have cookers, dishwashers, or washing machines. We will have robots to cook our dinner and wash our clothes. We will have robots to look after our children, and to feed our cats and dogs.
We won’t use computers. We will have special remote control units. We can surf the Internet, send and receive emails and order food from the markets.
We won’t have TVs or CD players because we will have watches that play music, take pictures and tell time. Our car might not use gas. They might use energy from air, water or plants. In the future the air won’t be polluted. Our world will be a great place to live in.
1. Our houses will use …………to make electricity.
A. only the wind B. only the sun
C. both the wind and the sun D. the wind or the sun
2. We won't have ……
A. robots B. cats and dogs C. energy D. dishwashers
3. We have ………….to do chores for us.
A. robots B. CD players C. cookers D. washing machines
4. We ………. with special remote control units.
A. look after our children B. feed our cats and dogs
C. order food from the markets D. take pictures
5. The passage is about…….
A. robots in the future B. new houses in the future
C. energy in the future D. chores in the future
XI. Rewrite the complete sentence without changing their original meanging.
1. The television is in front of the picture.
→ The picture is …………………………………………………………………
2. Put used paper in the recycling bins.
→ We should …………………………………..………………………..
3. There is an animated movie on at 7.15 this evening.
→ The animated movie starts ………………………………………...
4. It's a good idea if we go there by bus.
→ We …………………………………………….....................………………
5. It isn't necessary for him to go walking at night.
→ He doesn't …………………………………………………………………
6.We live near school, but they don’t.
→ They live ………………………………………………………………….……..
7. My future house will probably have smart doors.
→ My future house might …………………………………………………………
8. We use fewer cars. We reduce pollution.
→ If we ……………………………………………………………………….
9. The clinic is behind the bank.
→ The bank ………………………………………………………………………
10. Why don't we go to school by bike every day.
→ Let's ………………………………………..………………………..
11. Trains are slower than planes.
→ Planes are ……………………………………………………………
12. Does the city have ten million people?
→ Are there………………………………………………………………………….?
13. We might not shop in the mall.
→ We will ………………………………………………………………….……..
14. We cycle to school every day. We keep fit.
→ If we ……………………………………………………………………….
15. The police station is to the left of the movie theater and the post office is to the right of the movie theater.
→ The movie theater …………………………………..………………………..
16. We shouldn't turn on the lights in the morning.
→ Don't ………………………………………..………………………..
17. It's good if you go to Sapa in summer.
→ You ……..…………………………………….....................………………
18. She is interested in watching television.
→ She likes …………………………………………………………………………
19.We will probably go to the moon for our summer holidays in 2050.
→ We might ………………………………………………………………….……..
20. We plant more trees. The air is fresher.
→ If we ……………………………………………………………………….
21. The dog is at the side of the cat.
→ The dog …...……..………………..……………………………………….…
22. Shall we turn off the tap when we aren't using water?
→ I suggest ..…………………………………..………………………..
23. It is necessary for her to go on a cycling tour.
→ She ………………………………………………………………….………
24. My favorite home is the eco-friendly home.
→ I like………………..……………………..………………….…………………..
25. People might use 3D printers to make things.
→ People will ……………………………………………………………….……..
26. The boy eats so many green apples. He is ill.
→ If the boy ………………………………………………………………….
27. I want to live in a house on the sea.
→ I would ………………..…………………………………...……………………..
28. We plan to see a movie next week.
→ We are ………………………………………………………………..……
29. Dalat is quieter than Saigon.
→ Saigon is ……………………………………………………………
30. I suggest meeting outside the movie theater at 7.30 tonight.
→ Let's ……….…………………………………………………..……
XII. Make sentences using the words given in brackets. Do not change the given words.
1. A lot of people will live in very large cities. (MEGACITIES)
→ A lot of people ……………………………………………………………..
2. You can join one of their cleanups. (PART)
→ You can ……………………………………………………..
3. What time is the movie on? (START)
→ What time………………………………………………………………….?
4. It is necessary for us to play sports. (NEED)
→ We ……………..……………………………………………………..
5. We can learn about sea life because we go to the aquarium. (SO)
→ We ……………..………………………………………………………
6. I want to live in a house that computer will control things in it. (SMART HOME)
→ I want ………………………………………………………………………..
7. Old people don't like science fiction movies. (INTERESTED)
→ Old people aren't ….……………………………………………………….
8. The students collect trash on the beach. (PICK)
→ The students ………………………………………………..
9. The train station is to the left of the cinema. (RIGHT)
→ The cinema ..…………………………………………………………………
10. There is a shopping mall facing the train station. (OPPOSITE)
→ There is ………………………………………………………………………
11. They look after the animals. (TAKE)
→ They ………………………………………………………..
12. Shall we meet at 7 on Friday evening? (ABOUT)
→ How ……………………………………………………………………….?
13. It's not a good idea if you hunt animals. (SHOULDN'T)
→ You ……………..………………………….…………………………
14. They want to build a big apartment building under the ground.(EARTHSCRAPER)
→ They want …………………………………………………………………..
15. The pen is 20.000 dong , the pencil is 12.000 dong. (EXPENSIVE)
→ The pen …………………………………………..………………………
16. Sorry, could you repeat that? (SAY)
→ Sorry, could you …………………………………………..?
17. Which cartoon do you like better? (PREFER)
→ Which cartoon …………………………………………………………….?
18. Why don't we visit local buildings? (SUGGEST)
→ I ……………..…………………………………………………………
19. The lake is to the right of her house and the hospital is to the left of her house. (BETWEEN)
→ Her house………………………………………………………………………
20. The English book is 50.000 dong. The Vietnamese book is 20.000 dong. (CHEAP)
→ The Vietnamese book is …………………………………………………
21. The cinema is at the side of the arcade. (NEXT)
→ The cinema …………………………………………………………………..
22. We think a few people will live in houses under the sea. (UNDERWATER)
→ We think ……………………………………………………………………..
23. Next to my house there is a bookshop. (BESIDE)
→ There is……………………………………………………………………….
24. I am going to the movie theater with my sister tonight. (ARE)
→ I and my sister ...….……………………………………………………….
25. You can offer to help or give money. (DONATE)
→ You can ……………………………………………………..
làm đi hộ tui nhanh còn làm cái khác
Tham khảo:
Để chứng minh \( QM + QD < AM + AD \), chúng ta có thể sử dụng bất đẳng thức tam giác. Trong trường hợp này, \( QM \) và \( QD \) là độ dài các đoạn thẳng, nên chúng ta có thể áp dụng bất đẳng thức tam giác để chứng minh điều cần chứng minh.
Bất đẳng thức tam giác cho biết rằng trong một tam giác bất kỳ, tổng độ dài của hai cạnh bất kỳ phải lớn hơn độ dài cạnh còn lại. Áp dụng bất đẳng thức tam giác vào tam giác \( AMD \), ta có:
\[
AM + AD > MD
\]
Tương tự, áp dụng bất đẳng thức tam giác vào tam giác \( QMD \), ta có:
\[
QM + QD > MD
\]
Kết hợp hai bất đẳng thức trên, ta có:
\[
(QM + QD) + (AM + AD) > 2 \times MD
\]
Nhưng vì \( Q \) nằm trong tam giác \( AMD \), nên \( MD \) không lớn hơn \( MA \) (vì \( Q \) nằm trong tam giác \( AMD \), nên \( MD \) không vượt quá \( MA \)). Vì vậy:
\[
2 \times MD < MA + AD
\]
Tổng hợp lại, ta có:
\[
(QM + QD) + (AM + AD) > MA + AD
\]
Tức là:
\[
QM + QD > AM + AD
\]
Vậy, đã chứng minh được \( QM + QD < AM + AD \).