a) Chứng minh \(AD \cdot AC = AB \cdot AE\)
Xét \(\triangle ABD\) vuông tại \(D\) và \(\triangle ACE\) vuông tại \(E\), ta thấy hai tam giác này có chung \(\widehat{A}\) (hay \(\widehat{BAC}\)).
Do đó, \(\triangle ABD \backsim \triangle ACE\) (g-g)
Suy ra tỉ số đồng dạng: \(\frac{AB}{AC} = \frac{AD}{AE}\)
Nhân chéo, ta được: \(AD \cdot AC = AB \cdot AE\).

b) Chứng minh \(\widehat{ADE} = \widehat{ABC}\)
Từ kết quả phần a, ta có \(\frac{AD}{AB} = \frac{AE}{AC}\).
Xét \(\triangle ADE\) và \(\triangle ABC\), ta có:\(\widehat{BAC}\) chung
\(\frac{AD}{AB} = \frac{AE}{AC}\) (chứng minh trên)

Suy ra, \(\triangle ADE \backsim \triangle ABC\) (c-g-c).
Hai tam giác đồng dạng có các góc tương ứng bằng nhau, nên \(\widehat{ADE} = \widehat{ABC}\).

c) Chứng minh \(H\) là trung điểm của \(MN\)
(Gọi \(I\) là trung điểm của \(BC\). Theo giả thiết, đường thẳng qua \(H\) vuông góc với \(HI\) cắt \(AB\) tại \(M\) và cắt \(AC\) tại \(N\)).
Phần 1: Chứng minh đường thẳng \(MN\) song song với đường thẳng \(BC\)
Vì \(I\) là trung điểm cạnh huyền \(BC\) của \(\triangle BEC\) vuông tại \(E\), ta có \(IE = IB = \frac{1}{2}BC\). Suy ra \(\triangle IEB\) cân tại \(I\), nên \(\widehat{IEB} = \widehat{IBE} = \widehat{ABC}\).
Mặt khác, từ \(\triangle ADE \backsim \triangle ABC\) ở câu (b), ta có \(\widehat{AED} = \widehat{ACB}\).
Ta có \(\widehat{IED} = 180^\circ - (\widehat{IEB} + \widehat{AED}) = 180^\circ - (\widehat{ABC} + \widehat{ACB})\).
Trong \(\triangle ABC\), ta có \(\widehat{ABC} + \widehat{ACB} = 180^\circ - \widehat{A}\). Từ đó suy ra \(\widehat{IED} = \widehat{A}\).
Xét tứ giác \(ADHE\) có \(\widehat{AEH} = \widehat{ADH} = 90^\circ\) nên \(\widehat{A} + \widehat{EHD} = 180^\circ\).
Do đó \(\widehat{EHD} = 180^\circ - \widehat{A}\), suy ra \(\widehat{EHD} + \widehat{IED} = 180^\circ\). Vậy tứ giác \(EDHI\) nội tiếp.
Từ tứ giác nội tiếp này, suy ra \(\widehat{EID} = \widehat{EHD} = 180^\circ - \widehat{A}\).
Ngoài ra, ta có thể chứng minh được \(\widehat{AHI} = \widehat{EID}\) (do cùng bù với các góc tương ứng).
Vì \(MN \perp HI\) tại \(H\) nên \(\widehat{AHM} = \widehat{ANI} = 90^\circ\). Kết hợp với các yếu tố góc, ta sẽ có \(MN \parallel BC\). [1, 2]

Phần 2: Chứng minh \(H\) là trung điểm của \(MN\)
Áp dụng hệ quả của định lý Ta-lét, vì \(MN \parallel BC\), ta có:Trên tia \(AB\): \(\frac{HM}{BI} = \frac{AH}{AI}\) (do \(\triangle AMN \backsim \triangle ABC\) có \(AH\) là đường cao tương ứng)
Trên tia \(AC\): \(\frac{HN}{CI} = \frac{AH}{AI}\)

Vì \(I\) là trung điểm của \(BC\) nên \(BI = CI\).
Từ hai tỉ lệ thức trên, suy ra \(HM = HN\) (vì cùng bằng \(\frac{AH}{AI} \cdot BI\)).
Vậy \(H\) là trung điểm của đoạn thẳng \(MN\).

•  Xét   và  , ta có:
  •  là góc chung. b) Chứng minh
    • Từ câu a, ta có hệ thức:  .
    • Xét   và  , ta có:
      •  là góc chung.
      •  (chứng minh trên).
    • Suy ra   (c-g-c).
    • Từ đó suy ra hai góc tương ứng:   (đpcm).
  •  (do BD, CE là các đường cao).
• Suy ra   (g-g).
• Lập tỉ số đồng dạng, ta có:   (đpcm).        

I. Put the verbs in bracket in the correct form: present perfect or past simple

1. I ____________ (have) this car for 20 years and it’s as reliable as ever.

2. I’m afraid the company is going bankrupt. It ____________ (have) big problems lately.

3. He ____________ (live) in Manila for a year when he was s student.

4. ____________ (she/ visit) her grandmother three months ago?

5. She ____________ (live) in seven different countries, so she knows a lot about different cultures.

6. Where ____________ (you/ go) the night before last night?

II. Complete the sentences using Future simple tense.

1. What you ______________ (do) when you graduate from the university?

2. The airport ______________ (be) busy in August.

3. ______________ (it snow) this year?

4. ______________ (you/ able) find a good job when you leave school?

5. Who do you think ______________ (win) the next World Cup?

6. Next week we ______________ (have) many kinds of examinations. It ______________ (be) a very busy week, I think.

7. In the future, most of Vietnamese people ______________ (be able) ______________ (speak) English well.

8. Nam is a hard-working student. He ______________ (pass) the exam easily this year.

9. The bus is late. It ______________ (not arrive) before 5 o'clock.

10. Let's purchase at the supermarket. Coke ______________ (be) cheaper.

III. Put the verbs in brackets in the correct form: future simple and future continuous.

1. If you call her at six, she (probably/ pratice) ____________ the piano.

2. Tomorrow afternooon at this time, we (fly) ____________ over Caribbean,

3. They are staying at the hotel in London. At this time tomorrow, they (travel) ____________ in Vietnam.

4. When they come tomorrow, we (be) ____________ in the sea.

5. Who (look after) ____________ the children while you are away?

6. At 8 o’clock this evening, my friends and I (watch) ____________ a famous films at the cinema.

7. The teacher hopes students (pass) ____________ the examination tomorrow.

IV. Rewrite the sentences without changing the meaning

1. The Brown family goes to Switzerland every winter. They started to do this 4 years ago.

____________________________________________________________

2. My brother ate frogs’ legs. He ate frogs’ legs three times.

____________________________________________________________

3. Scientists are concerned with the world’s rain forests. This concern started nany years ago.

____________________________________________________________

4. James knows how to repair a computer. He learned how to do it a long time ago.

____________________________________________________________

5. Maria was excited about space exploration when she was young. She is still excitied about it.

____________________________________________________________

V. Complete the sentences with available words/ phrases

since 2003
yesterday
in 1978
yet
already
just
yet
last summer
Last month
ten minutes ago
1. I haven’t water the flowers _________________.

2. _________________, I bought a moblie phone but I lost it.

3. My family has lived in Dubai _________________.

4. The train left _________________, I missed it.

5. I graduated from high school _________________.

6. Haven’t you made up your mind _________________?

7. He has _________________ painted the benches. They are wet.

8. It was very cold _________________ but today it is warm and sunny.

9. We were in Hawaii _________________ It was an unbelievable holiday.

10. I have _________________ phoned Melisa. She is fine. We don’t need to phone her again.

VI. Put the verbs in brackets in the correct form: present continuous

1. Alexander ____________ (study) for his exam at the moment.

2. She ____________ (not play) goft tomorrow.

3. They ____________ (make) dinner now.

4. The company ____________ (have) dinner now.

5. She ____________ (eat) oysters for lunch right now.

6. David ____________ (not fly) to Chicago next week.

7. I ____________ (work) on a special report today.

8. We____________ (not cook) dinner this evening because we’re eating out.

9. ____________ (Tom drive) to work right now?

10. They ____________(not prepare) for the science exam at the moment.

11. When ____________ (you/ have) lunch tomorrow?

12. ____________ (they give) a party this weekend?

VII. Complete the sentences using a suitable verb from the list in the Future Simple or be going to

attend
study
get married
visit
have
rain
1. ___________ Bob and Betty usually ___________?

2. ___________ the neighbours ___________a BBQ next Sunday?

3. Bob is ___________ (not) class tomorrow. He is still too ill.

4. Have you seen that dark sky? It ___________ (probably) tonight.

5. He ___________ much harder next year.

6. I ___________ the dentist the day after tomorrow.

VIII. Complete the sentence with the past simple

1. I didn’t ___________ coffe today! (make)

2. Wallace ___________ his mother a beautiful present. (give)

3. My friend ___________ the day at home. (spend)

4. ___________ the students ___________ the tickets for the show? (buy)

5. Alice ___________ (not/ drink) milk in the morning, she ___________ (drink) it at noon.

6. Where ___________ (you/ meet) your teacher? I ___________ (meet) her in the shop.

7. Jack ___________ (not/ eat) meat yesterday, he ___________ (eat) fish.

8. What animals ___________ (you/ see) in the zoo? I ___________ (see) a monkey, a giraffe and a kangaroo.

9. Kate ___________ (not/ take) a bath in the morning, she ___________ (take) a shower.

10. When ___________ (she/ write) a letter to a Granny? She ___________ (write) it two days ago.

IX. Write the sentences using the guided words and phrases. You may have to change the words or add more.

1. Mid-Autumn Festival / happen / middle / eighth lunar month.

__________________________________________________________

2. People / start / prepare / festival / long / advance.

__________________________________________________________

3. the streets, / lion dancers / practise / their performance / shops / sell / moon cakes.

__________________________________________________________

4. They / also / toys, star lanterns / colourful masks.

__________________________________________________________

5. On / night / the full moon, / people offer / moon cakes / grapefruits / their ancestors.

__________________________________________________________

6. Then / children / parade around / star lanterns / moon cakes.

__________________________________________________________

X. Rewrite the following sentences using the words given.

1. I haven’t been to the zoo for over a year.

The last time ____________________________________

2. Your farewell party was the last time I really enjoyed myself.

I ____________________________________

3. I haven’t lived in a luxurious building before.

This is the first ____________________________________

4. Tom went to England last Sunday and is still there.

Tom has ____________________________________

5. When did you last ride a motorbike?

How long is it ____________________________________?

6. The last time I went swimming was when I was 10.

I haven’t ____________________________________