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Chapter 10: Problem 78

If a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle and an isosceles right triangle have the same perimeter, which will have the greater area? Why?

Short Answer

Expert verified
The 30掳-60掳-90掳 triangle has a greater area.

Step by step solution

01

Understand the Perimeters

Identify the properties of a 30掳-60掳-90掳 triangle and an isosceles right triangle. For a 30掳-60掳-90掳 triangle, the sides are in the ratio 1:鈭�3:2, and for an isosceles right triangle, the sides are in the ratio 1:1:鈭�2.
02

Define the Perimeters

Let the sides of the 30掳-60掳-90掳 triangle be a, a鈭�3, and 2a. The perimeter is a + a鈭�3 + 2a = a(3 + 鈭�3). Let the sides of the isosceles right triangle be b, b, and b鈭�2. The perimeter is b + b + b鈭�2 = b(2 + 鈭�2).
03

Equalize the Perimeters

Set the perimeters of both triangles equal to each other: a(3 + 鈭�3) = b(2 + 鈭�2). Simplify to find a in terms of b, or vice versa.
04

Calculate the Areas

For the 30掳-60掳-90掳 triangle, the area is (1/2)ab, with b = a鈭�3. For the isosceles right triangle, the area is (1/2)b^2. Express the areas using the sides in terms of a single variable.
05

Simplify the Area Comparison

Use the relationship between a and b from step 3 to compare the areas of the two triangles.
06

Conclusion

Determine which triangle has the greater area based on the simplified expressions from step 5.

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Key Concepts

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

30-60-90 Triangle
A 30掳-60掳-90掳 triangle is a special right triangle with unique properties. The angles are always 30 degrees, 60 degrees, and 90 degrees. This triangle has sides in a specific ratio: 1 : 鈭�3 : 2. This means:
  • The shortest side (opposite the 30掳 angle) is 'a'.
  • The side opposite the 60掳 angle is 'a鈭�3'.
  • The hypotenuse (opposite the 90掳 angle) is '2a'.
These side ratios help us quickly identify and solve problems involving a 30掳-60掳-90掳 triangle. For example, if you know one side length, you can easily find the others using these ratios.
Isosceles Right Triangle
An isosceles right triangle is another special kind of triangle. It has a right angle (90 degrees) and two equal angles of 45 degrees each. The sides of an isosceles right triangle are in the ratio 1 : 1 : 鈭�2. This means:
  • Each leg (the sides forming the right angle) is 'b'.
  • The hypotenuse (the side opposite the right angle) is 'b鈭�2'.
Since the legs are equal, this simplifies many calculations. For example, the perimeter of an isosceles right triangle is simply 2b + b鈭�2. This property makes them very useful in geometry problems.
Perimeter
The perimeter of a triangle is the sum of its side lengths. For the 30掳-60掳-90掳 triangle with sides 'a', 'a鈭�3', and '2a', the perimeter is:a + a鈭�3 + 2a = a(3 + 鈭�3).
  • For the isosceles right triangle with sides 'b', 'b', and 'b鈭�2', the perimeter is:b + b + b鈭�2 = b(2 + 鈭�2).When these perimeters are equal, we set up the equation:
  • a(3 + 鈭�3) = b(2 + 鈭�2).
    • From here, we solve for 'a' in terms of 'b' to understand their relationship better. This relationship allows us to compare different properties like area.
Area Calculation
The area of a triangle is given by (1/2) base 脳 height.For the 30掳-60掳-90掳 triangle:Area = (1/2) * a * a鈭�3 = (a^2鈭�3)/2.For the isosceles right triangle:Area = (1/2) * b * b = (b^2)/2.To compare the areas, we use the relationship between 'a' and 'b' derived from the perimeter equality:a(3 + 鈭�3) = b(2 + 鈭�2).From this, we get 'a' in terms of 'b' or vice versa. Then: Places these expressions in the area formulas, simplifying and comparing, shows which has the larger area.

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Most popular questions from this chapter

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