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Chapter 11: Problem 14

One triangle has sides \(13,13,\) and \(10 .\) A second triangle has sides \(12,20,\) and \(16 .\) Find the ratio of their areas.

Short Answer

Expert verified
The ratio of the area of the first triangle to the area of the second triangle is the answer obtained after dividing the area of the first triangle by the area of the second.

Step by step solution

01

Find the Area of the First Triangle

Use Heron's formula to calculate the area of the first triangle. Heron's formula is: \(A=\sqrt{s(s-a)(s-b)(s-c)}\) where \(s\) is the semi-perimeter of the triangle and \(a, b, c\) are the sides of the triangle. For the first triangle, \(a=b=13\) and \(c=10\) and \(s=\frac{a+b+c}{2}\). So first calculate \(s\), then substitute the values into Heron's formula to get the area of the first triangle.
02

Find the Area of the Second Triangle

Use Heron's formula to calculate the area of the second triangle. For the second triangle, \(a=12, b=20,\) and \(c=16\) So first calculate \(s\) for the second triangle, then substitute the values into Heron's formula to get the area of the second triangle.
03

Compare the Areas

Divide the area of the first triangle by the area of the second triangle to get their ratio. Ensure you simplify the ratio to its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Area
Finding the area of a triangle can seem challenging, but with Heron's Formula, it鈥檚 quite manageable. Heron's Formula is especially useful when you know the lengths of all three sides and need to calculate the area without having the height. The formula is given by \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) represents the semi-perimeter of the triangle, and \(a, b, c\) are the lengths of the sides. By plugging the values into the formula, you can uncover the exact area effortlessly! This method is powerful because it gives you the ability to calculate triangular areas even when standard base-height pairs are unknown.
Semi-perimeter
The semi-perimeter is a crucial part of Heron's formula. It makes calculating the area of a triangle straightforward. The semi-perimeter \(s\) is half of the perimeter of the triangle. You find it by adding all the side lengths and dividing by two: \(s = \frac{a+b+c}{2}\). This value balances the formula and is central to calculating the area accurately. Finding the semi-perimeter is simple yet essential, transforming complex area calculations into an organized process.
  • For the first triangle with sides 13, 13, and 10: \(s = \frac{13+13+10}{2} = 18\)
  • For the second triangle with sides 12, 20, and 16: \(s = \frac{12+20+16}{2} = 24\)
Area Ratio
Once you have the areas of both triangles, finding the area ratio is straightforward. The area ratio compares the sizes of the two areas, providing a deeper insight into their relative dimensions. It鈥檚 calculated by dividing the area of one triangle by the other. Ensure the ratio is simplified to its simplest form for clarity and accuracy. For instance, if the first triangle's area is smaller than the second, the ratio might be something like 1:2. This succinctly tells us that the second triangle鈥檚 area is twice as large as the first.
Geometry Problem Solving
Geometry problems often require a strategic approach for effective solutions. Breaking them into smaller, manageable steps can simplify the process. Start by determining what is known and what needs to be found. Use formulas effectively, ensuring you understand each component, like the semi-perimeter in Heron's formula.
Visualize the problem if possible鈥攄rawing a diagram can greatly aid comprehension. Once you have solutions, check them by re-evaluating calculations and considering alternative methods. This methodological approach not only helps in solving complex problems but also deepens your understanding of geometric principles.

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Most popular questions from this chapter

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