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

Redeem

One side of a triangle has lenght 8 and a second side...

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Legendary Member
Posts: 2499
Joined: Sun Oct 29, 2017 2:04 pm
Followed by:6 members

One side of a triangle has lenght 8 and a second side...

by swerve » Tue Dec 05, 2017 10:59 am
One side of a triangle has lenght 8 and a second side has lenght 5. Which of the following could be the area of the triangle?

I. 24
II. 20
III. 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.
Top
Source: — Problem Solving |
Master | Next Rank: 500 Posts
Posts: 100
Joined: Wed Nov 29, 2017 4:38 pm
Thanked: 14 times

by GMATWisdom » Tue Dec 05, 2017 1:10 pm
when only two sides of a triangle are given
the minimum value of area is zero and
the maximum area is half of the product of the two given sides.
so in our case the minimum area would be zero and the maximum would be

(1/2) x 8 x 5 = 20.

hence D is the correct answer.

[As per Trigonometry area of a triangle is given as

= (1/2) abSin(x)
where a and b are the lengths of the sides of a triangle and (x) is the angle between them.
so area is minimum and equal to zero when (x) is zero degree. and
the area is maximum = (1/2) x 8 x 5 = 20 when (x) = 90 degrees]

let me know if you are satisfied
Top
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Tue Dec 05, 2017 5:23 pm
Let's avoid trig, since the GMAT doesn't expect us to know it.

Visualizing the triangle, we can make the area about as small as we'd like by simply extending the third side as far as it can go:

Image

As our base approaches 8 + 5, our height approaches 0, so any area slightly north of 0 is possible. That means our lower bound is 0: we can't have 0, but we can have anything a hair away from it.

The upper bound is straightforward enough: we'll get it when the sides we have (5 and 8) are the legs of a right triangle. We can visualize this too:

Image

See how the height (the orange line) shrinks when we bent the blue and red legs off a right angle?

That gives us a maximum of blue = height, red = base, or 5 * 8 / 2, or 20.

Summing up, we're left with 0 < area ≤ 20, and from there the problem solves itself.
Top
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Tue Dec 05, 2017 5:24 pm
Also, for those flashcarding this:

Given two legs of a triangle, the area of the triangle has the range: 0 < area ≤ (given side * other given side) / 2
Top
Post new topic Post Reply
• Page 1 of 1