Finding The Height Of A Triangle: A Geometry Deep Dive

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Finding the Height of a Triangle: A Geometry Deep Dive

Hey everyone! Today, we're diving headfirst into a classic geometry problem. We're gonna figure out how to find the height of a triangle when we know some key information: two sides, and the radius of the circle that goes around the whole triangle (the circumcircle). The sides we're dealing with are 8 cm and 15 cm, and the radius of the circumcircle is 10 cm. Ready to get started? Let's break it down step-by-step to make sure we understand everything perfectly. This problem is a fantastic way to flex our geometry muscles and remember some cool formulas. Also, this type of problem is super common in tests and exams, so understanding it will definitely come in handy!

Understanding the Problem: The Setup

Okay, so the problem gives us a triangle. We know that two of its sides are 8 cm and 15 cm long. We also know something about the circle that can be drawn around this triangle – its radius is 10 cm. Remember that the circle goes around the outside of the triangle, touching all three corners. This is called the circumcircle, and its center is called the circumcenter. What we need to find is the height of the triangle drawn to the third side. The third side is the one we don’t directly know the length of. The height is the perpendicular distance from the top corner (vertex) opposite the third side, down to that side. It forms a right angle. Thinking visually is key in geometry. Try to sketch out the triangle with the 8 cm and 15 cm sides, then draw the circumcircle around it. Mark the radius (10 cm) wherever you can. This simple drawing can help us understand the problem. Before we jump into calculations, let's also remember what the area of a triangle is. The basic formula is (1/2) * base * height. If we can find the area, and we know the length of the third side (our base), then we can calculate the height. This is the path we'll try to follow. Knowing the circumradius gives us an extra tool that we can use, and we will do so with the Law of Sines.

Visualize the Triangle and Circle

Imagine that we have a triangle with sides of 8 cm and 15 cm. A circle envelops it, touching all three vertices. The radius of this circle is 10 cm. The height we are looking for is a line segment from one of the vertices to the opposite side, forming a right angle with that side. This opposite side will be our base for calculating the area. Visualizing this helps to understand the relationships between the sides, angles, and the circumcircle. It's like having a map of what's happening. And as we continue with the steps, we will be able to confirm if our reasoning and drawing are correct. This visual approach will make the whole process easier to understand.

Key Formulas and Concepts

To crack this problem, we need to bring out some important formulas. Don't worry, they're not too complicated, and you can easily remember them with a bit of practice. Let's list some that will be very important for us.

  1. Area of a triangle: The basic one, Area = (1/2) * base * height. We can use this to find the height if we know the area and the base.
  2. Law of Sines: For any triangle, the ratio of a side's length to the sine of its opposite angle is constant and is equal to twice the radius of the circumcircle. This is a huge deal. It looks like this: a / sin(A) = b / sin(B) = c / sin(C) = 2R, where a, b, and c are the side lengths, A, B, and C are the angles opposite those sides, and R is the radius of the circumcircle.
  3. Heron's Formula: This formula helps us find the area of a triangle when we know the lengths of all three sides. First, find the semi-perimeter (s) which is half the perimeter of the triangle: s = (a + b + c) / 2. Then, the area (A) is calculated as A = sqrt(s * (s - a) * (s - b) * (s - c)).
  4. Area using sides and circumradius: The area of a triangle can also be calculated as: Area = (abc) / (4R), where a, b, and c are the sides of the triangle and R is the circumradius.

Why These Formulas Matter

These formulas are like the keys to unlock our problem. The area formula is directly linked to what we want to find – the height. The Law of Sines is essential because it links the sides of a triangle to the angles and the circumradius, which we have. Heron's formula is important for calculating the area when we know the lengths of all the sides. If we can find the area using Heron's Formula, and we know the length of the base, we can easily find the height. Finally, the formula relating sides, and the circumradius allows us to cross-check our calculations and ensure we are on the right track. Each formula acts like a piece of the puzzle, and by using them together, we can get to the final answer. So, take a moment to understand each one, because they’ll be our best friends in this geometry adventure.

Step-by-Step Solution

Alright, buckle up, guys! We're now going to work through the solution step by step. We'll start with the information we have and, bit by bit, figure out the height.

Step 1: Find the length of the third side using the area formula

We know two sides (8 cm and 15 cm) and the circumradius (10 cm). We can use the formula: Area = (abc) / (4R), where a = 8 cm, b = 15 cm, and R = 10 cm. The third side, c, is what we need to find.

First, let's rearrange the formula to find the third side: c = (4 * Area * R) / (ab). But we need to find the area first. We can do this using the Law of Sines and the known sides and the circumradius. Let A be the area of our triangle. We can use the following formula. A = (abc) / (4R) => A = (8 * 15 * c) / (4 * 10) => A = (120c) / 40 => A = 3c. We need to use another formula to find the area A. We know that the area of the triangle can be calculated using the following formula: A = 0.5 * a * b * sin(C) , where C is the angle between sides a and b. We can use the Law of Sines to find the value of sin(C). From the Law of Sines we can get: c / sin(C) = 2R => c / sin(C) = 20 => sin(C) = c / 20. Then we can replace it into the area formula: A = 0.5 * 8 * 15 * c / 20 => A = 60c / 20 => A = 3c. The area is also equal to the area calculated above, so that is another way to check our calculations. Now we need to find the area using Heron's formula. But we don’t know all three sides. We need to find the third side. Here, we're going to use a neat trick. Let's rearrange the formula: A = (abc) / (4R) to find the area directly. This helps us find the area without needing the third side initially. Since we know A = 3c we can also calculate the area with the following formula: A = 0.5 * a * b * sin(C). This gives us a new formula A = 0.5 * 8 * 15 * sin(C) => A = 60 * sin(C). And as we saw before with the Law of Sines: c/sin(C) = 2R, then c = 2R * sin(C) => c = 20 * sin(C). From A = 3c and c = 20 * sin(C) then we get: A = 3 * 20 * sin(C) => A = 60 * sin(C) which is also the same as we got previously. Let's solve for the area, knowing that A = 3c. We will try to find all sides of the triangle and then calculate its area using Heron's formula.

Step 2: Use the Law of Cosines to calculate the third side

The Law of Cosines is our next stop. This law connects the lengths of the sides of a triangle to the cosine of one of its angles. It states that: c^2 = a^2 + b^2 - 2ab * cos(C). Where c is the third side, a and b are the known sides (8 cm and 15 cm), and C is the angle opposite side c.

To use this, we need to find the angle C. We'll use the Law of Sines, again! We know that c / sin(C) = 2R. But we don't have the value of c. So, we'll try something else. We can also use another formula for calculating the area: A = 0.5 * a * b * sin(C). And we also know that Area = 3c. So we can use them together: 3c = 0.5 * 8 * 15 * sin(C) => 3c = 60 * sin(C) => c = 20 * sin(C). Also, from the Law of Sines: c / sin(C) = 2R = 20 => c = 20 * sin(C). So, now we have two equations with two variables: c and sin(C). However, we can also use the Law of Cosines. c^2 = a^2 + b^2 - 2ab * cos(C). Now, we know that sin^2(C) + cos^2(C) = 1. We also know that A = 60 * sin(C) (from step 1). Let's work with the equation c = 20 * sin(C), and the law of cosines: c^2 = 8^2 + 15^2 - 2 * 8 * 15 * cos(C) => c^2 = 64 + 225 - 240 * cos(C) => c^2 = 289 - 240 * cos(C). Now, we have two variables here: c and cos(C). We know that sin^2(C) + cos^2(C) = 1, from which we get: cos(C) = sqrt(1 - sin^2(C)). From the Law of Sines, we know that c / sin(C) = 20. Then, sin(C) = c / 20. Replacing it into cos(C), we get: cos(C) = sqrt(1 - (c/20)^2). Replacing it into the Law of Cosines: c^2 = 289 - 240 * sqrt(1 - (c/20)^2) => c^2 - 289 = -240 * sqrt(1 - (c^2 / 400)). We can square both sides: (c^2 - 289)^2 = 240^2 * (1 - c^2/400). => c^4 - 578c^2 + 83521 = 57600 - 144c^2 => c^4 - 434c^2 + 25921 = 0. This is a quadratic equation with respect to c^2. Using the quadratic formula, we get: c^2 = (434 +- sqrt(434^2 - 4 * 25921)) / 2 => c^2 = (434 +- sqrt(188356 - 103684)) / 2 => c^2 = (434 +- sqrt(84672)) / 2 => c^2 = (434 +- 291) / 2. Then we get two results for c^2 : c^2 = 362.5 and c^2 = 71.5. And for c: c = 19.03 cm and c = 8.46 cm. These are two possible solutions for the length of the third side.

Step 3: Calculating the area with Heron's formula

Now, we'll calculate the area using Heron's formula. We have two sets of sides: (8 cm, 15 cm, 19.03 cm) and (8 cm, 15 cm, 8.46 cm). We will calculate the area for both of them.

Case 1: Sides are 8 cm, 15 cm, 19.03 cm

First, find the semi-perimeter: s = (8 + 15 + 19.03) / 2 = 21.015 Area = sqrt(21.015 * (21.015 - 8) * (21.015 - 15) * (21.015 - 19.03)) => Area = sqrt(21.015 * 13.015 * 6.015 * 1.985) = sqrt(3266.3) = 57.15 cm^2.

Case 2: Sides are 8 cm, 15 cm, 8.46 cm

First, find the semi-perimeter: s = (8 + 15 + 8.46) / 2 = 15.73 Area = sqrt(15.73 * (15.73 - 8) * (15.73 - 15) * (15.73 - 8.46)) => Area = sqrt(15.73 * 7.73 * 0.73 * 7.27) = sqrt(660.8) = 25.71 cm^2.

Step 4: Finding the Height

Now we can find the height using the formula Area = (1/2) * base * height. We have two areas and two possible base lengths, so we'll calculate the height for each case.

Case 1: Area = 57.15 cm^2, base = 19.03 cm

Height = (2 * Area) / base = (2 * 57.15) / 19.03 = 6.00 cm.

Case 2: Area = 25.71 cm^2, base = 8.46 cm

Height = (2 * Area) / base = (2 * 25.71) / 8.46 = 6.07 cm.

Step 5: Final Answer

So, there are two possible heights for the triangle. Based on the given information, we have two possible solutions for the height. Depending on which solution is suitable, the height of the triangle is approximately 6.00 cm or 6.07 cm. The variation can be determined by the angle of the triangle and the position of the height. But in general both solutions are correct.

Conclusion: Geometry is Awesome!

That was a fantastic journey through the world of triangles, guys! We started with a tricky problem and, step by step, used our knowledge of geometry to find the height. We used formulas like the Law of Sines, the Law of Cosines, the basic area formula, and Heron's formula. By breaking the problem down and using these tools, we found the solution. Always remember, the key to geometry is to understand the relationships between different parts of a figure. Keep practicing and keep exploring, and you'll be amazed at what you can solve. If you found this helpful, let me know. Do you want to try another geometry problem next time? Let me know in the comments. Thanks for sticking around; until next time! Keep learning, keep exploring, and keep having fun with math.