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In the figure above, if BD is perpendicular to AC and AC = 21, then AD

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Joined: 02 Sep 2009
Posts: 53798
In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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New post 25 Feb 2019, 02:46
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A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

89% (01:15) correct 11% (02:09) wrong based on 27 sessions

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In the figure above, if BD is perpendicular to AC and AC = 21, then AD =

A. 12
B. 13
C. 20
D. 21
E. 25

Attachment:
2019-02-25_1345.png
2019-02-25_1345.png [ 18.14 KiB | Viewed 309 times ]

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Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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New post 25 Feb 2019, 03:01
Bunuel wrote:
Image
In the figure above, if BD is perpendicular to AC and AC = 21, then AD =

A. 12
B. 13
C. 20
D. 21
E. 25

Attachment:
2019-02-25_1345.png



\(BD^2 = 13^2 - 5^2\)

\(BD^2 = 144\)

BD = 12.

Given ,

AC = 21

BC = 5

AB = 16.

\(AD^2 = BD^2 + AB^2\)

\(AD^2 = 12^2 + 16^2\)

\(AD^2 = 144 + 256\)

\(AD^2 = 400\)

AD = 20.

C is the correct answer.
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Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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New post 25 Feb 2019, 20:39
Triangle BCD is a right angle triangle
thus \(BD^2\)=\(CD^2\)-\(BC^2\)
hence \(13^2\)-\(5^2\) =144
BD=12

now considering other right angle triangle ABD
\(BD^2+ AB^2 = AD^2\)
\(12^2 +(21-5)^2 = 12^2+16^2 = 400\)
thus AD =20

Answer C
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Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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New post 25 Feb 2019, 22:22

Solution


Given:
    • In the given figure, BD is perpendicular to AC
    • AC = 21

To find:
    • The length of AD

Approach and Working:
As BD is perpendicular to AC, we can say triangle DBC and triangle DBA are both right-angled triangles.

In triangle DBC,
    • DC = 13 and BC = 5
    • Therefore, from Pythagoras Theorem, we can say DB = \(\sqrt{13^2 – 5^2}\) = 12

In triangle DBA,
    • DB = 12 and BA = 21 – 5 = 16
    • Therefore, from Pythagoras Theorem, we can say AD = \(\sqrt{12^2 + 16^2}\) = 20

Hence, the correct answer is option C.

Answer: C

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Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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New post 26 Feb 2019, 01:15
Bunuel wrote:
Image
In the figure above, if BD is perpendicular to AC and AC = 21, then AD =

A. 12
B. 13
C. 20
D. 21
E. 25

Attachment:
2019-02-25_1345.png


Use pythagorean triplets:

Triangle BCD is a right triangle. Since hypotenuse is 13 and one leg is 5, the other leg BD = 12 (since 5-12-13 is a pythagorean triplet)

Triangle ABD is also a right triangle such that AB = 21 - 5 = 16 and BD = 12. The two legs are 12 and 16 (in the ratio 3:4). So this is a multiple of pythagorean triplet 3-4-5 with a multiplier of 4. The hypotenuse AD will be 5*4 = 20

Answer (C)
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Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD   [#permalink] 26 Feb 2019, 01:15
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In the figure above, if BD is perpendicular to AC and AC = 21, then AD

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