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In triangle ABC, if BC = 3 and AC = 4, then what is the

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In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 01 Feb 2012, 15:55
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Triangle.jpg [ 8.62 KiB | Viewed 10293 times ]
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?

A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
[Reveal] Spoiler: OA

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Last edited by Bunuel on 19 Dec 2012, 01:41, edited 2 times in total.
Edited the question and added the figure
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 26 Feb 2012, 08:42
Hi

I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?

So here BC should be the perpendicular bisector and the AC=CD=3 right ?

Please let me know what am missing here.

I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 26 Feb 2012, 11:07
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rvinodhini wrote:
Hi

I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?

So here BC should be the perpendicular bisector and the AC=CD=3 right ?

Please let me know what am missing here.

I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?


A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.

Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
splittingtriangle.jpg [ 4.22 KiB | Viewed 14257 times ]

Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).

So, \frac{CD}{AC}=\frac{AC}{BC} --> \frac{CD}{4}=\frac{4}{3} --> CD=\frac{16}{3}.

Answer: D.

For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 26 Feb 2012, 20:09
Now I get it ..wonderful explanation !!
thanks
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 27 Feb 2012, 08:11
thankx for the explanation..
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In triangle ABC to the right, if BC = 3 and AC = 4, then [#permalink] New post 21 Apr 2012, 08:26
MBAhereIcome wrote:
all three triangles abc, acd & abd are similar.
so, \frac{4}{3} = \frac{cd}{4} ... cd = \frac{16}{3}

it can't be \frac{4}{3} = \frac{4}{cd} ... cd = 3, because in that case 5^2+5^2 = 6^2 is not true


If the 3 triangles are proportional, why can't I solve using the ratio:

\frac{AD}{AB} = \frac{AB}{(X+3)}
\frac{4}{5} = \frac{5}{(X+3)}
4(X+3) = 25
4X + 12 = 25
X = \frac{13}{4} :?
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then [#permalink] New post 21 Apr 2012, 09:35
Expert's post
pubchum wrote:
MBAhereIcome wrote:
all three triangles abc, acd & abd are similar.
so, \frac{4}{3} = \frac{cd}{4} ... cd = \frac{16}{3}

it can't be \frac{4}{3} = \frac{4}{cd} ... cd = 3, because in that case 5^2+5^2 = 6^2 is not true


If the 3 triangles are proportional, why can't I solve using the ratio:

\frac{AD}{AB} = \frac{AB}{(X+3)}
\frac{4}{5} = \frac{5}{(X+3)}
4(X+3) = 25
4X + 12 = 25
X = \frac{13}{4} :?

Attachment:
splittingtriangle.jpg
splittingtriangle.jpg [ 4.22 KiB | Viewed 12216 times ]

In similar triangles the ratio of the corresponding sides are equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).

The ratios you are equating are not of corresponding sides. If you want to equate AD/AB then it should be AD/AB=AC/BC --> AD/5=4/3 --> AD=20/3. Also AD/AB=CD/AC --> (20/3)/5=CD/4 --> CD=16/3.

I merged this thread with an earlier discussion of the same question, so check this post: in-triangle-abc-if-bc-3-and-ac-4-then-what-is-the-126937.html#p1050155 it might help to clear your doubts.

Please ask if anything remains unclear.
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 17 May 2012, 04:46
Hi Bunuel,

How did you infer that angle BAC = angle ADC. Could you please explain that part.. I know this can be proved thro similarity, however I wanted to understand from your concept, which you have mentioned below.
Pls explain.

Thanks
H
Bunuel wrote:
rvinodhini wrote:
Hi

I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?

So here BC should be the perpendicular bisector and the AC=CD=3 right ?

Please let me know what am missing here.

I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?


A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.

Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg

Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).

So, \frac{CD}{AC}=\frac{AC}{BC} --> \frac{CD}{4}=\frac{4}{3} --> CD=\frac{16}{3}.

Answer: D.

For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.

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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 17 May 2012, 04:59
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Expert's post
imhimanshu wrote:
Hi Bunuel,

How did you infer that angle BAC = angle ADC. Could you please explain that part.. I know this can be proved thro similarity, however I wanted to understand from your concept, which you have mentioned below.
Pls explain.

Thanks
H


Attachment:
splittingtriangle.jpg
splittingtriangle.jpg [ 4.22 KiB | Viewed 11937 times ]

<B+<D+<A=180, since <A=90 then <B+<D=90;
Similarly in triangle ABC: <B+<BAC=90 since <B=90-<D then (90-<D)+<BAC=90 --> <BAC=<D.

Hope it's clear.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 17 May 2012, 06:36
Thanks for the solution. I just wanted to know whether it requires any calculation or is it just a corollary of the property described above. Its clear now.

Bunuel wrote:
imhimanshu wrote:
Hi Bunuel,

How did you infer that angle BAC = angle ADC. Could you please explain that part.. I know this can be proved thro similarity, however I wanted to understand from your concept, which you have mentioned below.
Pls explain.

Thanks
H


Attachment:
splittingtriangle.jpg

<B+<D+<A=180, since <A=90 then <B+<D=90;
Similarly in triangle ABC: <B+<BAC=90 since <B=90-<D then (90-<D)+<BAC=90 --> <BAC=<D.

Hope it's clear.

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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 20 Jun 2013, 05:39
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COLLECTION OF QUESTIONS:
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 20 Jun 2013, 06:18
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Well i have a problem with similar triangles coz i sometimes make mistakes on the common sides. hence alternative approach for this problem..

Considering Triangle ACD - AC^2 + CD^2 = AD^2

Considering Triangle ABD - AB = sqrt(4^2 + 3^2) = 5 (Pythagorean triplet so you dont really have to do the math on the test)

Considering Triangle BAD - AB^2 + AD^2 = BD^2

25 + ac^2 + cd^2 = (3 + cd)^2

=> 25+16+cd^2= 9 + 6cd +cd^2

=>32 = 6cd

cd = 16/3
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 22 Jun 2013, 07:23
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enigma123 wrote:
Attachment:
Triangle.jpg
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?

A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3



Learn this property:

AC^2=BC*DC

=>DC=16/3
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Re: In triangle ABC, if BC = 3 and AC = 4, then what is the [#permalink] New post 22 Jun 2013, 23:00
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In the triangles ABD, <CAB= <CDA
----> tan <CAB = tan <CD
----> 3/4=4/CD
----> CD=16/3
--->Answer D
Re: In triangle ABC, if BC = 3 and AC = 4, then what is the   [#permalink] 22 Jun 2013, 23:00
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