Question

In Exercises $21-28,$ decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. $$ a=23, \quad b=19, \quad c=12 $$

Name: Problem 24, Chapter 5: Precalculus: Graphical, Numerical, Algebraic
Uploaded: 2019-07-18T04:00:40-08:00
Duration: 1 min 51 s
Channel: Qi Tong Liu
Description: In Exercises 21-28, decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. a=23, b=19, c=12

   In Exercises $21-28,$ decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle.
$$
a=23, \quad b=19, \quad c=12
$$

Precalculus: Graphical, Numerical, Algebraic

Franklin D.… 8th Edition

Chapter 5, Problem 24

Instant Answer

Solved by Expert Qi Tong Liu

07/18/2019

Step 1

According to the triangle inequality theorem, a triangle can be formed if the sum of the lengths of any two sides is greater than the length of the third side. So, we check if $a+b>c$, $b+c>a$, and $a+c>b$. Here, $23+19=42>12$, $19+12=31>23$, and $23+12=35>19$. Show more…

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In Exercises $21-28,$ decide whether a triangle can be formed with the given side lengths. If so, use Heron's Formula to find the area of the triangle. $$ a=23, \quad b=19, \quad c=12 $$

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Key Concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry that states that for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This condition must be satisfied for all three combinations of added side lengths, ensuring that the sides can connect to enclose a space.

Heron's Formula

Heron's Formula is a method for calculating the area of a triangle when all three side lengths are known. It involves first finding the semiperimeter, which is half the sum of the side lengths, and then using the formula that takes the square root of the product of the semiperimeter and the semiperimeter minus each side length. This approach allows for the computation of the area without needing to know the height of the triangle.

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