Tóm tắt lý thuyết Toán 8 Bài 3: Hình thang – Hình thang cân ngắn gọn, chính xác sách Chân trời sáng tạo sẽ giúp học sinh nắm vững kiến thức trọng tâm, ôn luyện để học tốt Toán 8.
Lý thuyết Toán lớp 8 Bài 3: Hình thang – Hình thang cân
A. Lý thuyết Hình thang – Hình thang cân
1. Khái niệm
Hình thang là tứ giác có hai cạnh đối song song.
Hình thang cân là hình thang có hai góc kề một đáy bằng nhau.
2. Tính chất của hình thang cân
+ Hai cạnh bên bằng nhau.
+ Hai đường chéo bằng nhau.
Ví dụ:
Hình thang cân EFGH có hai cạnh bên EH = FG, hai đường chéo EG = FH.
3. Dấu hiệu nhận biết hình thang cân
- Hình thang có hai góc kề một đáy bằng nhau là hình thang cân.
- Hình thang có hai đường chéo bằng nhau là hình thang cân.
B. Bài tập Hình thang – Hình thang cân
Bài 1. Cho hình thang cân ABCD (AB // CD). Hai tia phân giác của A và B cắt nhau tại điểm K nằm trên cạnh đáy DC. Từ K kẻ đoạn thẳng KM vuông góc với AB tại M.
a) Chứng minh ∆ABK là tam giác cân.
b) Chứng minh AM = BM.
Hướng dẫn giải
a) Do AK là tia phân giác của Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span nên Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
BK là tia phân giác của Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span nên Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Mà ABCD là hình thang cân nên Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span (tính chất hình thang cân).
Do đó Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , suy ra ∆ABK là tam giác cân tại K.
b) Vì ∆ABK là tam giác cân nên KM là đường cao và cũng là đường trung tuyến ứng với cạnh đáy AB.
Do vậy M là trung điểm của AB nên AM = BM.
Bài 2. Tính số đo các góc chưa biết của hình thang IJGH (IJ // GH) trong các trường hợp sau:
a) ˆ H = 50 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , ˆ J = 110 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
b) IJGH là hình thang cân và ˆ H = 30 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
c) ˆ H = 90 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , ˆ G = 50 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Hướng dẫn giải
a)
Do IJ // GH nên ta có:
ˆ I + ˆ H = 180 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , suy ra ˆ I = 180 ° − ˆ H = 180 ° − 50 ° = 130 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
ˆ J + ˆ G = 180 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , suy ra ˆ G = 180 ° − ˆ J = 180 ° − 110 ° = 70 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
Vậy hình thang IJGH có: ˆ I = 130 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ; ˆ G = 70 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
b)
Hình thang cân IJGH (IJ // GH) có ˆ I = ˆ J Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span và ˆ G = ˆ H = 30 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span (tính chất hình thang cân).
Áp dụng định lí tổng các góc của một tứ giác, ta có:
ˆ I + ˆ J + ˆ G + ˆ H = 360 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
Suy ra 2 ˆ I = 360 ° − 2 ˆ H = 360 ° − 2.30 ° = 300 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
Do đó ˆ I = ˆ J = 150 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Vậy hình thang cân IJGH có: ˆ G = 30 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ; ˆ I = ˆ J = 150 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
c)
ˆ H = 90 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span nên hình thang IJGH là hình thang vuông, suy ra ˆ H = ˆ I = 90 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
Áp dụng định lí tổng các góc của một tứ giác, ta có:
ˆ J = 360 ° − ( ˆ I + ˆ G + ˆ H ) = 360 ° − ( 90 ° + 50 ° + 90 ° ) = 360 ° − 230 ° = 130 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ⎛ ⎝ Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ⎞ ⎠ Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ( Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ) Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span
Vậy hình thang vuông IJGH có: ˆ I = 90 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span , ˆ J = 130 ° Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Bài 3. Cho tam giác cân EFG có EF = EG. Trên các cạnh EF và EG, lần lượt lấy các điểm H và I sao cho EH = EI.
a) Chứng minh HIGF là hình thang.
b) Chứng minh HIGF là hình thang cân.
Hướng dẫn giải
Ta có EH = EI nên ∆EHI là tam giác cân tại E.
Suy ra Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span (1)
Lại có EF = EG nên ∆EFG là tam giác cân tại E.
Suy ra Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span (2)
Từ (1) và (2) suy ra Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Mà Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span và Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span ở vị trí đồng vị nên HI // FG (dấu hiệu nhận biết).
Suy ra tứ giác HIGF là hình thang.
b) Vì ∆EFG là tam giác cân nên Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Suy ra hình thang HIGF là hình thang cân.
Bài 4. Cho hình thang ABCD (AD // BC) có AC = BD.
a) Hình thang ABCD là hình thang gì? Vì sao?
b) Chứng minh Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span .
Hướng dẫn giải
a) Vì hai đường chéo của hình thang ABCD là AC và BD bằng nhau nên hình thang ABCD là hình thang cân.
b) Theo tính chất hình thang cân, ta có: AB = DC
Xét ∆ABD và ∆DCA có:
AD là cạnh chung;
AB = DC;
BD = AC.
Suy ra ∆ABD = ∆DCA (c.c.c)
Do đó Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span Unknown node type: span (hai cạnh tương ứng).
VIDEO
Video bài giảng Toán 8 Bài 3: Hình thang – Hình thang cân - Chân trời sáng tạo