**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}

½b_{1}h_{1} = ½b_{2}h_{2}

We know that b_{1} (base of ∆ABC) is bigger than b_{2} (base of ∆DEF), so let’s just choose some arbitrary numbers to make it a bit easier to see the relationship. Let’s let b_{1} = 6 and b_{2} = 4.

½(6)(h_{1}) = ½(4)(h_{2})

3h_{1} = 2h_{2}

h_{1} = ⅔h_{2}

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

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