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Giải bài tập 7.10 trang 38 SGK Toán 9 tập 2 - Cùng khám phá

Cho đường tròn tâm O có bán kính R = 5 cm. a) Tính độ dài cạnh của hình vuông nội tiếp trong (O). b) Một hình chữ nhật nội tiếp (O) có chu vi 28 cm. Tính chiều dài và chiều rộng của hình chữ nhật đó.

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Đề bài

Cho đường tròn tâm O có bán kính R = 5 cm.

a) Tính độ dà𓆉i cạnh của hình vuông n🔯ội tiếp trong (O).

b) Một hình chữ nhật nội tiếp (O) có chu vi 28 cm. Tính chiều dài và chi🅷ều rộng của hìnꦕh chữ nhật đó.

Phương pháp giải - Xem chi tiết

Đọc kĩ dữ kiện để vẽ hình Áp dụng định lý: Đường tròn ngoại tiếp hình vuông, hình chữ nhật có tâm là giao điểm của hai đường chéo và bán kính bằng nửa đường chéo. Sau đó áp dụng định lí pythagore để tính cạnh hình vuông, chiều dài, rộng hình chữ nhật.

Lời giải chi tiết

a)

Ta có BD = 2OB = 2R = 2.5 = 10 cm. Áp dụng định lý Pythagore trong tam giác vuông BAD ta có: \(A{D^2} + A{B^2} = 2A{D^2} = BD\) suy ra AD = \(\sqrt {\frac{{BD}}{2}}  = \sqrt {\frac{{10}}{2}}  = \sqrt 5 cm\). b)

Ta có AC = 2AE = 2R = 2.5 = 10 cm. Nửa chu vi hình chữ nhật là: 28 : 2 = 14 cm Gọi AD = x suy ra CD = 14 – x Áp dụng định lí Pythagore trong tam giác vuông ADC có: \(\begin{array}{l}A{D^2} + C{D^2} = A{C^2}\\{x^2} + {\left( {14 - x} \right)^2} = {10^2}\\{x^2} + 196 - 28x + {x^2} - 100 = 0\\2{x^2} - 28x + 96 = 0\end{array}\) Giải phương trình ta được AD = 6 cm và CD = 8 cm.

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