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

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Math Expert
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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25 Feb 2019, 02:46
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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 [ 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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25 Feb 2019, 03:01
Bunuel wrote:

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.

$$AD^2 = BD^2 + AB^2$$

$$AD^2 = 12^2 + 16^2$$

$$AD^2 = 144 + 256$$

$$AD^2 = 400$$

C is the correct answer.
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Joined: 28 Jul 2016
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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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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
$$BD^2+ AB^2 = AD^2$$
$$12^2 +(21-5)^2 = 12^2+16^2 = 400$$

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Joined: 04 Jan 2015
Posts: 2722
Re: In the figure above, if BD is perpendicular to AC and AC = 21, then AD  [#permalink]

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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.

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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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26 Feb 2019, 01:15
Bunuel wrote:

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

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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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