BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Inscribed triangle - semicircle

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Master | Next Rank: 500 Posts
Posts: 243
Joined: Sun Jul 12, 2009 7:12 am
Location: Dominican Republic
Thanked: 31 times
Followed by:2 members
GMAT Score:480

Inscribed triangle - semicircle

by MAAJ » Mon Jan 24, 2011 12:19 pm
From the graphic, I understand that triangle PQS is similar to PQR because they share xº and 90º, so the last angle must be equal. But I'm missing why is PQS similar to QSR, can someone help me out? (I know they both share 90º but not sure about the other angles) Thanks!
Image
"There's a difference between interest and commitment. When you're interested in doing something, you do it only when circumstance permit. When you're committed to something, you accept no excuses, only results."
Top
Source: — Data Sufficiency |
Legendary Member
Posts: 857
Joined: Wed Aug 25, 2010 1:36 am
Thanked: 56 times
Followed by:15 members

by AIM GMAT » Mon Jan 24, 2011 12:54 pm
MAAJ wrote:From the graphic, I understand that triangle PQS is similar to PQR because they share xº and 90º, so the last angle must be equal. But I'm missing why is PQS similar to QSR, can someone help me out? (I know they both share 90º but not sure about the other angles) Thanks!
Image
PQS is similar to PQR for the below reason :-

It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.

Hope this helps.
Thanks & Regards,
AIM GMAT
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Mon Jan 24, 2011 3:51 pm
Please see my explanation for the problem below.

Image

Triangle PQR is a right triangle. QS is a height drawn through the right angle to the opposite side.

In any right triangle, a height drawn through the right angle to the opposite side creates 3 similar triangles.

We can prove this by plugging in for the angles measurements. If ∠RPQ = 20, then ∠PQS = 70, since the sum of the 2 angles must be 90. This forces ∠RQS to be 20. This forces ∠QRS to be 70. The result is that all 3 triangles (PQS, RQS, and PQR) have the same combination of angles: 20-70-90. Thus, the 3 triangles are similar.

The corresponding sides of similar triangles must yield the same proportion.
In triangle PQS, the shorter leg = QS, the longer leg = PS = 16.
In triangle RQS, the shorter leg = RS = 9, the longer leg = QS.
Since shorter leg:longer leg must be the same for each triangle, we get:
QS/16 = 9/QS
(QS)² = 144
QS = 12.

Thus, in PQR, h = 12 and b = PS + SR = 16+9 = 25.
Area = 1/2*b*h = 1/2(25)(12) = 150.

The correct answer is D.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 243
Joined: Sun Jul 12, 2009 7:12 am
Location: Dominican Republic
Thanked: 31 times
Followed by:2 members
GMAT Score:480

by MAAJ » Tue Jan 25, 2011 4:22 pm
Wow, many thanks! now I get it... I was missing the part of picking numbers for angles, thank U!
"There's a difference between interest and commitment. When you're interested in doing something, you do it only when circumstance permit. When you're committed to something, you accept no excuses, only results."
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3835
Joined: Fri Apr 02, 2010 10:00 pm
Location: Milpitas, CA
Thanked: 1854 times
Followed by:523 members
GMAT Score:770

by Anurag@Gurome » Tue Jan 25, 2011 9:59 pm
MAAJ wrote:Image
Note that in triangle PQR, (xº + yº) = 90º
Now, in triangle PQS, angle PQS = 90º - xº = yº

In triangle PQS and QSR,
  • angle PSQ = angle QSR
    angle PQS = angle QRS = yº
Hence, they are similar.

Just remember this fact for right-angled triangle that when you draw a perpendicular from the vertex with the right-angle on the hypotenuse, the new smaller two right-angled triangles are similar to each other and also similar to the original one.
Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 406
Joined: Mon Jan 25, 2010 11:36 am
Location: Syracuse, NY
Thanked: 23 times
Followed by:4 members
GMAT Score:740

by tomada » Fri Jan 28, 2011 11:18 am
Mitch, in triangle RQS, we know that RS < QS based on the diagram? (i.e. QS appears to be longer)


GMATGuruNY wrote:Please see my explanation for the problem below.

Image

Triangle PQR is a right triangle. QS is a height drawn through the right angle to the opposite side.

In any right triangle, a height drawn through the right angle to the opposite side creates 3 similar triangles.

We can prove this by plugging in for the angles measurements. If ∠RPQ = 20, then ∠PQS = 70, since the sum of the 2 angles must be 90. This forces ∠RQS to be 20. This forces ∠QRS to be 70. The result is that all 3 triangles (PQS, RQS, and PQR) have the same combination of angles: 20-70-90. Thus, the 3 triangles are similar.

The corresponding sides of similar triangles must yield the same proportion.
In triangle PQS, the shorter leg = QS, the longer leg = PS = 16.
In triangle RQS, the shorter leg = RS = 9, the longer leg = QS.
Since shorter leg:longer leg must be the same for each triangle, we get:
QS/16 = 9/QS
(QS)² = 144
QS = 12.

Thus, in PQR, h = 12 and b = PS + SR = 16+9 = 25.
Area = 1/2*b*h = 1/2(25)(12) = 150.

The correct answer is D.
I'm really old, but I'll never be too old to become more educated.
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Fri Jan 28, 2011 11:45 am
tomada wrote:Mitch, in triangle RQS, we know that RS < QS based on the diagram? (i.e. QS appears to be longer)
We know that RS < QS based upon the information given. Here again is the figure, with variables assigned to the angles:

Image

Since all 3 triangles are similar, in each triangle we get the following proportion:

(leg opposite x):(leg opposite y) = (leg opposite x):(leg opposite y)

This gives us the proportion QS/16 = 9/QS.
Since QS=12 and RS=9, RS < QS.

Which leg is shorter doesn't really matter. Just be sure to set up the correct proportion, as shown above.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
Post new topic Post Reply
• Page 1 of 1