Tag Archive for 'algebra'

Example Problem #1

Example Geometry Problem

This is a fun problem.  The problem tells us that ∆ABC and ∆DEF have the same area.  It also tells us that AD > CF.  The problem is asking us about altitude, which is height.

From this second piece of information, we know that the base of ∆ABC is longer than the base of ∆DEF.  How do we know this?  From the picture, we see that the base of ∆ABC is composed of two segments: AD + DC.  The base of ∆DEF is also composed of two segments: DC + CF.  Both triangles share DC — this is the same for both triangles.  But we know that AD is bigger than CF, which means that AD + DC > DC + CF.  Thus, the base of triangle ∆ABC is larger.

Now, since we know that both triangles have the same area, but the base of ∆ABC is bigger, this means that the height of ∆DEF has to be bigger in order to compensate.  The answer should be B, but let’s prove it.

In math terms:

Area_∆ABC = Area_∆DEF
½b₁h₁ = ½b₂h₂

We know that b₁ (base of ∆ABC) is bigger than b₂ (base of ∆DEF), so let’s just choose some arbitrary numbers to make it a bit easier to see the relationship.  Let’s let b₁ = 6 and b₂ = 4.

½(6)(h₁) = ½(4)(h₂)
3h₁ = 2h₂
h₁ = ⅔h₂

Therefore, we know that the height of ∆ABC is smaller than the height of ∆DEF.  Thus, the answer is B.

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