The three sides of a triangle are 144, 165 and 219 respectively

General
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the three sides of a triangle are 144, 165 and 219 respectively

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Understanding Triangle Sides and Properties

To analyze a triangle with sides measuring 144, 165, and 219, we’ll look into various aspects such as whether it forms a valid triangle, calculate its perimeter, explore its area using Heron’s formula, and classify the type of triangle it is.

1. Valid Triangle Verification

For any set of three lengths to form a triangle, they must satisfy the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let’s verify this:

  1. Check 1: 144 + 165 > 219
    309 > 219 (True)

  2. Check 2: 144 + 219 > 165
    363 > 165 (True)

  3. Check 3: 165 + 219 > 144
    384 > 144 (True)

Since all three conditions satisfy the triangle inequality, these sides can indeed form a triangle.

2. Calculating the Perimeter

The perimeter of a triangle is the sum of its sides. For this triangle, the perimeter P is calculated as:

P = 144 + 165 + 219 = 528

The perimeter of this triangle is 528 units.

3. Calculating the Area Using Heron’s Formula

Heron’s formula is used to find the area of a triangle when the lengths of all three sides are known. It is given by:

A = \sqrt{s(s-a)(s-b)(s-c)}

Where s is the semi-perimeter of the triangle, calculated by:

s = \frac{a+b+c}{2} = \frac{144 + 165 + 219}{2} = 264

Let’s calculate the area step-by-step:

  • The semi-perimeter s is 264.
  • Heron’s formula becomes:
A = \sqrt{264(264-144)(264-165)(264-219)}

Simplifying inside the square root:

  • (264 - 144) = 120
  • (264 - 165) = 99
  • (264 - 219) = 45

So, the formula is now:

A = \sqrt{264 \times 120 \times 99 \times 45}

Calculating inside the square root:

A = \sqrt{264 \times 120 \times 99 \times 45} \approx \sqrt{14098560}

Finally, the area A:

A \approx 3754.12 \text{ square units}

4. Classification of the Triangle

Triangles can be classified based on their side lengths:

  • Equilateral Triangle: All sides are equal.
  • Isosceles Triangle: Two sides are equal.
  • Scalene Triangle: All sides are different.

In this case, since all the sides are of different lengths (144, 165, 219), it is a scalene triangle.

5. Type by Angles

To determine the type of triangle by its angles (acute, right, or obtuse), we can use the cosine rule or check the largest angle by using the Pythagorean Theorem (for a quick check of a right triangle):

A triangle is a right triangle if:

  • c^2 = a^2 + b^2

Check:

219^2 \stackrel{?}{=} 144^2 + 165^2

Calculating:

  • 144^2 = 20736
  • 165^2 = 27225
  • 219^2 = 47961

Check the sum of squares:

  • 144^2 + 165^2 = 20736 + 27225 = 47961

Since 219^2 = 144^2 + 165^2, this triangle is a right triangle.

Summary

These given sides form a valid triangle with the triangle’s perimeter equal to 528 units and the area approximately 3754.12 square units. This triangle is classified as a scalene and a right triangle because it fulfills the conditions of the Pythagorean Theorem. This intriguing blend of properties helps us understand and visualize the geometric shape formed by sides 144, 165, and 219.

If you have any further questions or need more details, feel free to ask!

@anonymous4