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F.4 Maths
發問:
A rectangle BDEF of length Xcm is inscribed in a right-angled triangle ABC, where AB=8 cm and BC = 6 cm. (a) show that the area of BDEF is (-3/4 X的2次 + 6X)cm (b) find the area of the largest rectangle that can be inscribed in triangle ABC. Thank you in advance!
最佳解答:
Q: A rectangle BDEF of length x cm is inscribed in a right-angled triangle ABC, where AB ＝ 8 cm and BC ＝ 6 cm. ( a ) show that the area of BDEF is ( - 3x2/4 ＋ 6x ) cm (b) find the area of the largest rectangle that can be inscribed in triangle ABC. Sol: ( a ) Suppose D ( E, whatever you want ) is on side AB, BD ＝ x cm and let y cm be the length of BE △ABC and △ADF are similar triangles therefore, 8 : ( 8 － x ) ＝ 6 : y 8y ＝ 48 － 6x y ＝ ( 48 － 6x ) / 8 ＝ ( 24 － 3x ) / 4 area rectangle BDEF ＝ xy ＝ x * ( 24 － 3x ) / 4 ＝ - 3x2/4 ＋ 6x ( b ) - 3x2/4 ＋ 6x ＝ - 3/4 ( x2 ＋ 8x ) ＝ - 3/4 ( x2 ＋ 8x ＋ 16 ) ＋ 12 ＝ - 3/4 ( x ＋ 4 )2 ＋ 12 ≧ 12 hence, the max. area is 12 cm2 Ans: ( a ) as shown as above ( b ) 12 cm2
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