GMAT Math : Acute / Obtuse Triangles

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #10 : Calculating The Height Of An Acute / Obtuse Triangle

Given:  with  and .

Construct the altitude of  from  to a point  on . What is the length of ?

Possible Answers:

Correct answer:

Explanation:

 is shown below, along with altitude ; note that  has been extended to a ray  to facilitate the location of the point 

Isosceles_2

Since an exterior angle of a triangle has as its measure the sum of those of its remote interior angles, 

By definition of an altitude,  is perpendicular to , making  a right triangle and  a 30-60-90 triangle. By the 30-60-90 Triangle Theorem, shorter leg  of  has half the length of hypotenuse —that is, half of 48, or 24; longer leg  has length  times this, or , which is the correct choice.

Example Question #61 : Acute / Obtuse Triangles

Given:  with  and .

Construct the altitude of  from  to a point  on . What is the length of ?

Possible Answers:

Correct answer:

Explanation:

 is shown below, along with altitude .

Isosceles

Since , and , by definition, is perpendicular to  is a 30-60-90 triangle. By the 30-60-90 Triangle Theorem, , as the shorter leg of , has half the length of hypotenuse ; this is half of 30, or 15.

Example Question #62 : Acute / Obtuse Triangles

Given:  with , construct two  altitudes of : one from  to a point  on , and another from  to a point  on . Which of the following is true of the relationship of the lengths of  and ?

Possible Answers:

The length of  is nine-sixteenths that of 

The length of  is three-fourths that of 

The length of  is two-thirds that of 

The length of  is four-ninths that of 

The length of  is twice that of 

Correct answer:

The length of  is three-fourths that of 

Explanation:

The area of a triangle is one half the product of the length of any base and its corresponding height; this is , but it is also . Set these equal, and note the following:

That is, the length of  is three fourths that of that of 

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