Triangle ABC: Find BD Length

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Hey guys! Today, we're diving into a fun geometry problem involving triangle ABC. We're given the lengths of all three sides: AB = 23 cm, AC = 7 cm, and BC = 24 cm. The altitude from vertex A meets side BC at point D, and it splits the altitude into a 11:1 ratio. Our mission? To find the length of the segment BD. Let's break it down step by step!

Understanding the Problem

Before we jump into calculations, let's visualize what we're dealing with. We have a triangle that isn't necessarily a right triangle, and we're dropping a perpendicular line from vertex A to side BC. This line, AD, is our altitude, and it's crucial because it forms right angles with BC, which will be super helpful when we start using the Pythagorean theorem. Understanding that the altitude is divided into a 11:1 ratio means that if we denote the shorter segment as x, the longer segment is 11x. Their sum gives us the total length of the altitude AD, but we don't know that length yet. The point D divides the base BC into two segments, BD and DC, and it's the length of BD that we're trying to find. We will use the properties of right triangles formed by the altitude to relate the lengths of the sides and eventually solve for BD. Remembering basic geometry principles, such as the Pythagorean theorem and properties of similar triangles, will be very useful in tackling this problem. Now, let's roll up our sleeves and get into the math!

Setting Up the Equations

Alright, let's get our hands dirty with some equations. Let's denote the length of BD as x. Consequently, the length of DC would be 24 - x since BC = 24 cm. Now, let AD be the altitude from vertex A to side BC, and let's call its length h. According to the problem, this altitude is divided in a 11:1 ratio. However, the problem states that the altitude is divided in this ratio, which seems unusual since point D is where the altitude meets BC. We should consider that the problem might have meant that a line from vertex A to side BC (which happens to be the altitude) is divided in the ratio, or there may be some information missing. For now, we will proceed assuming the altitude is what's divided, and we will adjust accordingly if this assumption proves incorrect.

Using the Pythagorean theorem on triangles ABD and ACD, we can set up two equations:

  1. In triangle ABD: AB² = AD² + BD² => 23² = h² + x²
  2. In triangle ACD: AC² = AD² + DC² => 7² = h² + (24 - x)²

Now we have a system of two equations with two unknowns (h and x). We can solve for h² in both equations and set them equal to each other. This will give us an equation in terms of x only, which we can then solve.

Solving for BD

Let's isolate h² in both equations:

  1. h² = 23² - x² = 529 - x²
  2. h² = 7² - (24 - x)² = 49 - (576 - 48x + x²) = -527 + 48x - x²

Now, set these two expressions for h² equal to each other:

529 - x² = -527 + 48x - x²

Notice that the x² terms cancel out, which simplifies things nicely:

529 = -527 + 48x

Now, solve for x:

48x = 529 + 527 48x = 1056 x = 1056 / 48 x = 22

So, we found that BD = x = 22 cm.

Double-Checking Our Work

Now that we have a value for BD, let's plug it back into our equations to make sure everything checks out. We found that BD = 22 cm, so DC = 24 - 22 = 2 cm.

Using the Pythagorean theorem:

In triangle ABD: 23² = AD² + 22² => 529 = AD² + 484 => AD² = 45 In triangle ACD: 7² = AD² + 2² => 49 = AD² + 4 => AD² = 45

Since AD² is the same in both equations, our value for BD seems to be correct!

Accounting for the 11:1 Ratio (Revised Solution)

Okay, so far we haven't used the information about the 11:1 ratio. This suggests that there might have been a misunderstanding, or that the altitude itself wasn't what was divided. Let's rethink this, guys! The problem states the altitude is divided into segments with a ratio of 11:1. Let's denote the point on AD that divides it as E, such that AE:ED = 11:1. However, this information seems extraneous since our initial approach worked out smoothly without it. But let's consider the possibility that the problem statement has an implicit condition or a slight twist that necessitates using this ratio.

Let's call AD = 12y (since it's divided into 11:1 ratio, so AE = 11y and ED = y). Now, we can rewrite our Pythagorean equations as:

  1. In triangle ABD: 23² = (12y)² + x² => 529 = 144y² + x²
  2. In triangle ACD: 7² = (12y)² + (24 - x)² => 49 = 144y² + (24 - x)²

Subtracting the second equation from the first, we get:

529 - 49 = x² - (24 - x)² 480 = x² - (576 - 48x + x²) 480 = 48x - 576 48x = 1056 x = 22

So, we still arrive at BD = 22 cm. The fact that the 11:1 ratio didn't change our answer might indicate that, in this specific problem setup, the altitude division is incidental or that we're missing a deeper geometric insight. If this were a more complex problem, this ratio might affect the angles or some other properties, but given the side lengths and the direct application of the Pythagorean theorem, it seems it doesn't alter the final result for BD.

Final Answer

After carefully setting up and solving the equations using the Pythagorean theorem, and double-checking our work, we found that the length of segment BD is 22 cm. Even after considering the 11:1 ratio in the altitude's division, the result remained consistent. Therefore, our final answer is:

BD = 22 cm

Geometry can be a fun puzzle, and this problem was no exception. Keep practicing, and you'll become a geometry pro in no time!