Triangle ABC Ratio Of Segments BD And CD Equals 3

Hey guys! Let's dive into an interesting geometry problem involving triangle ABC. We're given that the ratio of the measure of segment BD to the measure of segment CD is equal to 3. Also, the hypotenuse BC measures 16cm. Our mission is to determine the measures of the segments determined by the altitude on the hypotenuse and the measure of the altitude relative to the hypotenuse. Buckle up, it's gonna be a fun ride!

Understanding the Problem

Before we jump into calculations, let's make sure we understand the problem inside and out. We have a right triangle ABC, where the right angle is at A. The hypotenuse BC is 16cm long. There's a point D on BC such that the ratio BD:CD is 3:1. We need to find the lengths of segments BD and CD, which are created by the altitude from A to BC. Additionally, we need to find the length of this altitude itself. Visualizing the problem can be super helpful, so if you're a visual learner, try sketching the triangle as we go.

Setting Up the Ratios and Equations

In tackling geometry problems, especially those involving ratios, it's crucial to set up the relationships correctly. The prompt tells us that the ratio of segment BD to segment CD is 3 to 1, which we can write as BD/CD = 3. This is a key piece of information that we'll use to figure out the actual lengths of these segments. But how do we translate this ratio into something more tangible? This is where algebra steps in to help us. Let's denote the length of CD as x. If CD is x, and BD is three times the length of CD, then BD would be 3x. Now we have algebraic expressions for both BD and CD in terms of a single variable, x. This is a significant step forward because it simplifies our calculations and allows us to solve for unknowns more easily.

The Hypotenuse Connection

We also know that the entire length of the hypotenuse BC is 16cm. This is our second golden nugget of information. The hypotenuse BC is made up of two segments, BD and CD. This means that the sum of the lengths of BD and CD must equal the length of BC. We can express this as an equation: BD + CD = 16 cm. But we've already expressed BD and CD in terms of x. Substituting these expressions, we get 3x + x = 16 cm. This simplifies to 4x = 16 cm, which is a straightforward algebraic equation that we can easily solve for x. Once we find x, we'll know the length of CD, and we can then find the length of BD by multiplying x by 3. This is where our problem begins to unravel, as these segment lengths are fundamental to solving for the altitude.

A) Determining the Measures of Segments BD and CD

Now that we have our equation 4x = 16 cm, let's solve for x. Dividing both sides of the equation by 4, we get x = 4 cm. So, CD is 4 cm long. Since BD is 3 times CD, BD = 3 * 4 cm = 12 cm. Fantastic! We've found the measures of segments BD and CD. This is a significant step, as these values are crucial for finding the altitude.

Applying the Pythagorean Theorem

To find the length of the altitude, we can use the Pythagorean Theorem. Let's call the altitude from A to BC as h. This altitude divides the original triangle ABC into two smaller right triangles: triangle ABD and triangle ACD. In each of these triangles, we can apply the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is a cornerstone of geometry and will help us relate the altitude to the known segments BD and CD.

Setting up the Equations

Let's denote the sides AB and AC as y and z, respectively. In triangle ABD, we have AB² = BD² + h², which translates to y² = 12² + h². In triangle ACD, we have AC² = CD² + h², which translates to z² = 4² + h². Now we have two equations, each involving the altitude h and one of the other unknown sides y and z. But we're still missing one crucial piece of information: a relationship between y and z that allows us to eliminate one of these variables. This is where the properties of similar triangles come into play.

B) Finding the Measure of the Altitude Relative to the Hypotenuse

Here's where things get interesting. Remember, the altitude we drew from A to BC creates two smaller triangles that are similar to the original triangle ABC. This is a fundamental concept in geometry: when you draw an altitude in a right triangle, you create similar triangles. Similar triangles have the same shape but can be different sizes, and their corresponding sides are in proportion. This proportionality is the key to solving for the altitude. The similarity of these triangles gives us crucial proportional relationships between their sides, which we can then use to find the length of the altitude.

Harnessing Similar Triangles

Because triangles ABD and ABC are similar, and triangles ACD and ABC are similar, we can set up proportions involving their corresponding sides. A key relationship we can use comes directly from the properties of similar triangles and the altitude. In the original triangle ABC, the square of the altitude (h²) is equal to the product of the lengths of the segments it creates on the hypotenuse. Mathematically, this is expressed as h² = BD * CD. This equation is a shortcut that bypasses the need to calculate the lengths of the other sides (AB and AC) individually. It is derived directly from the similarity of triangles and is a powerful tool in solving problems like this.

Calculating the Altitude

Now, let's plug in the values we know. We found that BD = 12 cm and CD = 4 cm. So, h² = 12 cm * 4 cm = 48 cm². To find h, we take the square root of both sides: h = √48 cm. We can simplify √48 by factoring out the largest perfect square. Since 48 = 16 * 3, √48 = √(16 * 3) = √16 * √3 = 4√3 cm. Therefore, the measure of the altitude relative to the hypotenuse is 4√3 cm. Yay, we solved it!

Wrapping It Up

So, to recap, we've found that the segments BD and CD measure 12 cm and 4 cm, respectively, and the altitude from A to BC measures 4√3 cm. This problem beautifully illustrates how different geometric concepts—ratios, the Pythagorean Theorem, and similar triangles—come together to solve a complex problem. Keep practicing, and you'll become a geometry whiz in no time!

In triangle ABC, the ratio of the length of segment BD to the length of segment CD is 3. The hypotenuse BC measures 16cm. A) Find the lengths of segments BD and CD formed by the altitude to the hypotenuse. B) Determine the length of the altitude to the hypotenuse.