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# If ∠ D C B is a right angle, then A C =

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Manager
Joined: 01 May 2016
Posts: 72
Location: United States
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If ∠ D C B is a right angle, then A C =  [#permalink]

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11 Feb 2018, 21:16
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1
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Difficulty:

45% (medium)

Question Stats:

65% (02:01) correct 35% (02:29) wrong based on 66 sessions

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If ∠ D C B is a right angle, then A C =

a 7

b 3+3√3

c 8

d 4+3√2

e 3+4√2

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If ∠ D C B is a right angle, then A C =  [#permalink]

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11 Feb 2018, 21:50
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In Triangle DCB(right-angled triangle), $$DB^2 = (4\sqrt{2})^2 + (4\sqrt{2})^2 = 64$$ -> DB = $$\sqrt{64} = 8$$

Triangles DXA and AXB are similar(SAS), therefore DX = XB = 4

In Triangle DXA, Side $$AX = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$$
In Triangle DXC, Side $$XC = \sqrt{(4\sqrt2)^2 - 4^2} = \sqrt{16} = 4$$

Therefore, $$AC = AX + XC = 3+4$$ = 7(Option A)

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Joined: 15 Feb 2018
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Re: If ∠ D C B is a right angle, then A C =  [#permalink]

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20 Dec 2018, 14:52
All this the same as pushpitkc, therefore DX = XB = 4

Recognise triangle DXA as a 3, 4, 5 right angle triangle. Therefore AX=3
Recognise triangle DXC as a 45, 45, 90 right angle triangle (angle C has been bisected, X is 90º, so part of D is also 45º). Again, this is a triangle to recognise. Hypotenuse is 4√2, so legs are 4.

4+3=7

These two, plus the 30-60-90 right angle triangle ones that very much should be memorised
Re: If ∠ D C B is a right angle, then A C =   [#permalink] 20 Dec 2018, 14:52
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