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Please solve with explanation (high points)

Please solve with explanation (high points)-example-1
User Krubo
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1 Answer

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a) From the Pythagorean theorem, we have

AC² = AB² + BC² = a² + a² ⇒ AC = √2 a

and

AK² = AB² + BK² = a² + (2a)² ⇒ AK = √5 a

ABCD and BKLC are perpendicular, so AC and CL are perpendicular and ∆ACL is a right triangle. By the Pythagorean theorem, we have

AL² = AC² + CL² = (√2 a)² + (2a)² ⇒ AL = √6 a

Now,

AK² + LK² = (√5 a)² + a² = 6a² = (√6 a)² = AL²

which means ∆AKL satisfies the Pythagorean theorem and is a right triangle, so ∠AKL = 90°.

b) We know AC = √2 a and AK = √5 a. By the Pythagorean theorem, we find

CK² = BC² + BK² = a² + (2a)² ⇒ CK = √5 a

By the law of cosines, we have

AK² = AC² + CK² - 2 AC CK cos(∠AKC)

6a² = 2a² + 5a² - 2√10 a² cos(∠AKC)

cos(∠AKC) = 1/(2√10)

∠AKC = arccos(1/(2√10)) ≈ 80.9°

User Magnar
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