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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