In triangle ABC, is AC^2 = AB^2 + BC^2 ?

Question

(1) ∠BAC + ∠ACB = ∠ABC
(2) AB = BC

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rishab0507 · Answer

In triangle ABC, is AC^2 = AB^2 + BC^2 ?

(1) ∠BAC + ∠ACB = ∠ABC
(2) AB = BC

Question stem is asking us whether ABC is right angled triangle or isocles right ∠ed triangle.

Statement 1 : 
∠BAC and ∠ ACB can be a 30-60 and ∠ ABC 90, forming 30-60-90 triangle, which will fulfill pythgores theorem, 
or can be 45-45-90 triangle which can fulfill pythagores theorem. So we get 2 answers,yes
Sufficient

Statement 2 : 
As we are looking for 3-4-5 or 6-8-10 etc type triangles, if AB= BC, then we can't fulfill pythagores theo. It can be 45-45-90 or 30-30- 120 triangles : So answer will be no and yes
In Sufficient

IMO A

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • ABC is a triangle. (Shown below)
  https://www.dropbox.com/s/uzgqxv6frylqoga/27_right.png?dl=1 
• We need to find if  AC^2 = AB^2+ BC^2 
 o The above equation represents the Pythagoras theorem for right angle.
  So basically, we need to find out whether triangle ABC is right angle or not.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  ∠BAC + ∠ACB = ∠ABC
 • ∠BAC + ∠ACB + ∠ABC = 180°  
  ⟹ 2∠ABC = 180° 
 ⟹ ∠ABC = 90° 
• Since, one of the interior angles of ABC is 90°, therefore ABC is a right -angled triangle. 
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  AB = BC
 • This statement only tell us that triangle ABC is isosceles, or ∠BAC = ∠ACB
 o However, we cannot conclude that the third angle is 90 degrees or ABC is a right angled triangle. 
• For example, 
 o If ∠BAC = ∠ACB =35 degrees ⟹ ∠ABC = 180 – 70 = 110 degrees
  Then ABC is not right angled triangle. 
o If ∠BAC = ∠ACB =45 degrees ⟹ ∠ABC = 180 – 90 = 90 degrees
  Then ABC is a right angled triangle.   
Hence, statement 2 is not sufficient.
Thus, the correct answer is  Option A.

suchita2409 · Answer

Gmat  WHIZ

From Statement 1 the 2 angles could be 45 45 .
For pythagoras theorem we need angles 30 60 90.

Am i missing sumthng

Posted from my mobile device

sambitspm · Answer

See the attachment
Answer A

Lipun · Answer

Any right angled triangle will satisfy Pythagoras theorem.

GMATWhizTeam · Answer

Hey suchita2409 The Pythagoras theorem is valid for all right angled triangles be it 45 -45 -90, 30-60-90, 40- 50-90, or any other combination of interior angles of a right angled triangle. The right angled triangle need not be only 30 - 60-90 triangle to apply this theorem. To understand it in a better way, let us consider the following two triangles. Triangle ABC is a 45 -45 -90 triangle and PQR is a 30 -60 -90 triangle.(As shown below) https://www.dropbox.com/s/ddulqj431g6sfck/pytha_proof.png?dl=1 Now, According to Pythagoras theorem, in a right angled triangle, (Hypotenuse)^2 = (Base)^2 + (Height)^2 • If we see in triangle ABC, Hypotenuse = x\sqrt{2} , height = base = x o (x\sqrt{2})^2 = x^2 +x^2 ⟹ 2x^2 = 2x^2 , this means Pythagoras theorem is valid in this case. • Now, if we see in triangle PQR, Hypotenuse = 2 , height = \sqrt{3} , and base = 1 o In this case also, 2^2 = (\sqrt{3})^2 +1^2 ⟹ 4 = 4 Thus, we can see that Pythagoras theorem is valid in both cases. I hope it helps.

Regards, GMATwhiz Team

In triangle ABC, is AC^2 = AB^2 + BC^2 ?

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