Geometry

[Aman verma]

Q:In the given triangle ABC, CD,BF,and AE are the altitudes. If the ratio of CD:AE:BF=2:3:4 then the ratio of AB:BC:CA is:

(A) 4:3:2

(B) 2:3:4

(C) 4:9:6

(D) 6:4:3

(E) 5:2:1

[theCodeToGMAT]

Is the answer [spoiler]{D}[/spoiler]?

[Mathsbuddy]

Area of triangle = base x perpendicular height/2

We are given the ratios of the heights and are asked for the corresponding ratios of the bases.

So, area * 2 = AB * CD = AC * BF = CB * AE

Substituting the given ratios gives:

2AB = 3AC = 4BC

So AC = 2AB/3 AND BC = AB/2

this means that AB:AC:BC = AB:2AB/3:AB/2

Multiplying by 6 and dividing by AB gives the simplified ratio of 6:4:3

Hence the answer is (D) 6:4:3

[GMATGuruNY]

Q:In the given triangle ABC, CD,BF,and AE are the altitudes. If the ratio of CD:AE:BF=2:3:4 then the ratio of AB:BC:CA is:

(A) 4:3:2

(B) 2:3:4

(C) 4:9:6

(D) 6:4:3

(E) 5:2:1

Any side of a triangle can be deemed the base.
Each base has a corresponding height.
Definition of height: the perpendicular distance between the base and the opposite vertex.

Let CD=2, AE=3, and BF=4.
Let the area = the LCM of 2, 3, and 4 = 12.

If AB is deemed the base, the corresponding height is CD.
Since the area = 12, we get:
(1/2)(AB)(CD) = 12
(1/2)(AB)(2) = 12
AB = 12.

If BC is deemed the base, the corresponding height is AE.
Since the area = 12, we get:
(1/2)(BC)(AE) = 12
(1/2)(BC)(3) = 12
BC = 8.

If AC is deemed the base, the corresponding height is BF:
Since the area = 12, we get:
(1/2)(AC)(BF) = 12
(1/2)(AC)(4) = 12
AC = 6.

Resulting ratio:
AB:BC:AC = 12:8:6 = 6:4:3.

The correct answer is D.