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2011-01-05 4:03 am

圖片參考：http://imgcld.yimg.com/8/n/HA00510450/o/701101040106513873436230.jpg

The figure show an isosceles triangle ABC with AB=AC and BC=10cm. AD⊥BC and AD=20cm. A rectangle PQRS is inscribed in △ABC. Suppose PQ=x cm.
a)express the area of rectangle PQRS in terms of x.
b)find the maximum area of rectangle PQRS and the corresponding dimensions.

更新1:

Why△APS ~ △ABC?

回答 (2)

用戶2053

2011-01-05 6:05 am

✔ 最佳答案

a)
△APS ~ △ABCSo (AD - PQ) / PS = AD / BC(20 - x) / PS = 20 / 10PS = (20 - x)/2The area of rectangle PQRS
= PQ * PS
= x * (20 - x)/2
= (20x - x²)/2
b)(20x - x²)/2
= - (x² - 20x) / 2
= - (x² - 20x + 100 - 100) / 2
= - [(x - 10)² - 100] / 2
= [- (x - 10)² + 100] / 2
When x = 10 ,
the maximum area of rectangle PQRS = (0 + 100)/2 = 50cm² ,
and the corresponding dimensions
= PQ * PS
= x * (20 - x)/2
= 10 * (20 - 10)/2
= 10cm * 5cm

2011-01-04 22:36:42 補充：
PS // BC since PQRS is a rectangle.

ㄥA = ㄥA (common)
ㄥS = ㄥC (corr. ∠s, PS//BC)
ㄥP = ㄥB (corr. ∠s, PS//BC)

△APS ~ △ABC (A.A.A.)

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[匿名]

2011-01-05 10:48 pm

Why△APS ~ △ABC?

∠PAS = ∠BAC (common side)
∠APS = ∠ABC (corr. ∠s ,PS=BC )
∠ASP = ∠ACB (corr. ∠s ,PS=BC )

∴ △APS ~ △ABC (equiangular )

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