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If a certain rectangle is inscribed in a triangle as shown above 30 Jan 2023, 05:08
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48% (02:33) correct 52% (02:26) wrong based on 33 sessions

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If a certain rectangle is inscribed in a triangle as shown above figure, what is the value of h, in terms of s and r?


A. \(\frac{\sqrt{3}*(s-r)}{2}\)

B. \(\frac{\sqrt{3}*(s+r)}{2}\)

C. \(\frac{\sqrt{3}*(s-r)}{2}\)

D. \(\sqrt{3}*(s-r)\)

E. \(\frac{\sqrt{3}*(s-r)}{4}\)

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If a certain rectangle is inscribed in a triangle as shown above 30 Jan 2023, 05:12
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Bunuel

If a certain rectangle is inscribed in a triangle as shown above figure, what is the value of h, in terms of s and r?


A. \(\frac{\sqrt{3}*(s-r)}{2}\)

B. \(\frac{\sqrt{3}*(s+r)}{2}\)

C. \(\frac{\sqrt{3}*(s-r)}{2}\)

D. \(\sqrt{3}*(s-r)\)

E. \(\frac{\sqrt{3}*(s-r)}{4}\)

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To the right bottom corner, we observe a 30-60-90 triangle in which side opposite to 60o is h

Side opposite to 30o \(= \frac{(s-r)}{2}\)

If side opposite to 30o is x then side opposite to 60o is x√3 as per 30o-60o-90o triangle property

hence, \(h = √3*\frac{(s-r)}{2}\)

Answer: Option C
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If a certain rectangle is inscribed in a triangle as shown above 30 Jan 2023, 14:13
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Hi,

Which side are you taking about when you say "If side opposite to 30º is x" ? ... Little lost here.. where is x ?

thks !!
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presc04
Hi,

Which side are you taking about when you say "If side opposite to 30º is x" ? ... Little lost here.. where is x ?

thks !!
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Hi presc04,

Please look at the attached fig. below:

As we know that angles of a \(\Delta\) add upto \(180.\)

So in case we have a \(\Delta\) in which the angles are \(30\) \(60 \) \(90\) then the sides obey a rule:

Based on this rule if we know the value one side of this \(30\) \(60 \) \(90 \) triangle we can find the value of the other two sides.

For example: suppose if side opposite to \(30\) angle is \(x\) then opposite \(60\) will be \(x\sqrt3\) and opposite \(90\) we will be \(2x \)

Attachment:
GMAT t 1.png
GMAT t 1.png [ 10.02 KiB | Viewed 1305 times ]

By the same rule if the side opposite \(60\) is \(h \) then side opposite \(30\) will be \(\frac{h}{\sqrt3}\) and side opposite \(90\) will be \(2h\)

Coming back to the original question:

Attachment:
GMAT t 2.jpg
GMAT t 2.jpg [ 8.08 KiB | Viewed 1304 times ]

Here also we have two \(\Delta \)'s in which the angles are \(30\) \(60\) and \(90\)

We can see how the side opposite to \(30\) is \(\frac{h}{\sqrt3} \) if the side opposite to \(60 \) is \(h\) as explained above.

Also the two \(\Delta\)'s in green are congruent so their corresponding sides are same.

so we can write s = \(\frac{h}{\sqrt3}+r +\frac{h}{\sqrt3}\)

\(\frac{2h}{\sqrt 3} = s-r\)

\(h= {\frac{(s-r)* \sqrt{3}}{2}\)

Let me know which part is still unclear.

( Also note : Answer choices A and C look same to me , probably a typo )
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If a certain rectangle is inscribed in a triangle as shown above 16 Feb 2023, 11:08
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The problem with this question is I can't tell what is referring to what in the figure. Is S the whole bottom of the triangle or just the segment that's part of the rectangle? Is h for "height" or "hypotenuse" of the smaller right triangle at the bottom right?
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If a certain rectangle is inscribed in a triangle as shown above 25 Feb 2023, 00:05
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Bunuel

If a certain rectangle is inscribed in a triangle as shown above figure, what is the value of h, in terms of s and r?


A. \(\frac{\sqrt{3}*(s-r)}{2}\)

B. \(\frac{\sqrt{3}*(s+r)}{2}\)

C. \(\frac{\sqrt{3}*(s-r)}{2}\)

D. \(\sqrt{3}*(s-r)\)

E. \(\frac{\sqrt{3}*(s-r)}{4}\)

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Bunuel which side is h?? please clarify this. Thanks in advance

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