Find Tan(B) In Triangle ABC: A Step-by-Step Solution

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Find tan(B) in Triangle ABC: A Step-by-Step Solution

Hey guys! Let's dive into a cool math problem today. We're going to tackle a triangle question that involves some geometry and trigonometry. Don't worry, we'll break it down step by step so it's super easy to follow. Our mission? To find the value of tan(B) in a given triangle. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we fully grasp the problem. We're given a triangle ABC with some specific properties. First off, sides AB and AC are equal in length, both measuring 13 cm. This immediately tells us we're dealing with an isosceles triangle, which is a triangle with two sides of equal length. The third side, BC, is 10 cm. We also have an angle relationship: angle BAC is twice the measure of angle B. Our ultimate goal is to determine the value of tan(B), which is the tangent of angle B. Remember, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

To recap, the key pieces of information we have are:

  • |AB| = |AC| = 13 cm (Isosceles Triangle)
  • |BC| = 10 cm
  • m(BAC) = 2 * m(B)
  • Goal: Find tan(B)

Now that we have a clear picture of the problem, let's start thinking about how we can use this information to find our answer. The fact that we have an isosceles triangle and an angle relationship gives us some clues. We'll need to use some geometric properties and trigonometric identities to solve this one. Let's move on to the solution strategy!

Strategy for Solving

Alright, so how are we going to crack this problem? Here's the plan of attack. Since we're dealing with an isosceles triangle and trying to find a trigonometric ratio, we'll need to combine geometric insights with trigonometric tools.

  1. Draw an Altitude: The first clever step is to draw an altitude from vertex A to side BC. Let's call the point where the altitude meets BC as D. In an isosceles triangle, this altitude has a special property: it bisects the base (BC) and the angle at the vertex (BAC). This means BD = DC, and angle BAD is equal to angle DAC.
  2. Use the Bisected Base: Since the altitude bisects BC, we know that BD = DC = BC / 2 = 10 cm / 2 = 5 cm. This gives us some concrete lengths to work with.
  3. Angle Relationships: We know that m(BAC) = 2 * m(B). Let's denote m(B) as θ (theta). Then m(BAC) = 2θ. Since the altitude AD bisects angle BAC, we have m(BAD) = m(DAC) = θ.
  4. Right Triangles: Drawing the altitude created two right triangles: triangle ABD and triangle ACD. We can use trigonometric ratios in these right triangles.
  5. Trigonometric Ratios: Focus on triangle ABD. We know the lengths of AB (13 cm) and BD (5 cm). We can use the Pythagorean theorem to find the length of AD. Then, we can use the definitions of sine, cosine, and tangent to express sin(θ), cos(θ), and tan(θ) in terms of the side lengths.
  6. Double Angle Identity: Remember that m(BAC) = 2θ. We can use the double angle identity for sine: sin(2θ) = 2 * sin(θ) * cos(θ). We can also express sin(2θ) in triangle ABC using the altitude. This will give us an equation that relates sin(θ) and cos(θ).
  7. Solve for tan(θ): Finally, we'll have enough information to solve for tan(θ), which is what we're looking for. We might need to use some algebraic manipulation and trigonometric identities to get there.

This strategy might seem like a lot of steps, but each step is manageable. By breaking the problem down into smaller parts, we can tackle it systematically. Let's start putting this strategy into action!

Step-by-Step Solution

Okay, let's roll up our sleeves and work through the solution step by step. Remember our strategy? We're going to draw an altitude, use the bisected base, leverage angle relationships, work with right triangles, apply trigonometric ratios, use the double angle identity, and finally, solve for tan(B). Let's get to it!

Step 1: Draw the Altitude

As planned, we draw an altitude AD from vertex A to side BC. This altitude creates two right triangles, ABD and ACD. Importantly, D is the midpoint of BC, and AD bisects angle BAC.

Step 2: Use the Bisected Base

Since D is the midpoint of BC, we know that BD = DC = BC / 2. Given that BC = 10 cm, we have BD = DC = 5 cm. This is a crucial piece of information for our calculations.

Step 3: Angle Relationships

We are given that m(BAC) = 2 * m(B). Let's denote m(B) as θ. Therefore, m(BAC) = 2θ. Since AD bisects angle BAC, we have m(BAD) = m(DAC) = θ. Now we have angles neatly defined in terms of θ.

Step 4: Right Triangles and Pythagorean Theorem

Focus on right triangle ABD. We know AB = 13 cm (the hypotenuse) and BD = 5 cm (one leg). We can use the Pythagorean theorem to find the length of AD (the other leg):

AB² = AD² + BD²

13² = AD² + 5²

169 = AD² + 25

AD² = 169 - 25

AD² = 144

AD = 12 cm

So, we now know AD = 12 cm. This is a major step forward!

Step 5: Trigonometric Ratios in Triangle ABD

In right triangle ABD, we can now express the trigonometric ratios for angle θ (angle B):

  • sin(θ) = Opposite / Hypotenuse = AD / AB = 12 / 13
  • cos(θ) = Adjacent / Hypotenuse = BD / AB = 5 / 13
  • tan(θ) = Opposite / Adjacent = AD / BD = 12 / 5

Wait a minute! We've already found tan(θ)! It's 12/5. But let's continue with our original plan and use the double angle identity just to confirm and reinforce our understanding.

Step 6: Double Angle Identity

We know that m(BAC) = 2θ. Let's consider triangle ABC. We can drop the altitude AD, which we already know is 12 cm. The area of triangle ABC can be calculated in two ways:

  1. Using base BC and height AD: Area = (1/2) * BC * AD = (1/2) * 10 cm * 12 cm = 60 cm²
  2. Using the formula: Area = (1/2) * AB * AC * sin(2θ) = (1/2) * 13 cm * 13 cm * sin(2θ)

Equating these two expressions for the area, we get:

60 cm² = (1/2) * 13 cm * 13 cm * sin(2θ)

60 = (1/2) * 169 * sin(2θ)

sin(2θ) = (2 * 60) / 169

sin(2θ) = 120 / 169

Now, let's use the double angle identity for sine: sin(2θ) = 2 * sin(θ) * cos(θ).

We already found sin(θ) = 12 / 13 and cos(θ) = 5 / 13. So,

sin(2θ) = 2 * (12 / 13) * (5 / 13)

sin(2θ) = 120 / 169

This matches our previous result for sin(2θ), which is a good sign!

Step 7: The Final Answer

As we found earlier in Step 5, tan(θ) = 12 / 5. So, the value of tan(B) is 12/5.

Conclusion

And there you have it! We've successfully navigated through this geometry and trigonometry problem. We started by understanding the givens: an isosceles triangle with specific side lengths and an angle relationship. Then, we developed a strategy that involved drawing an altitude, using the bisected base, leveraging angle relationships, working with right triangles, applying trigonometric ratios, and using the double angle identity. By systematically working through each step, we arrived at the answer: tan(B) = 12/5.

This problem showcases how combining geometric insights with trigonometric tools can help solve complex problems. The key is to break the problem down into manageable steps and use the properties of the shapes and functions involved. Great job, guys! You've tackled a challenging problem and come out on top. Keep practicing, and you'll become math whizzes in no time! Remember, practice makes perfect, and understanding the fundamentals is crucial for success in mathematics.