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Chapter 5: Problem 42

Explain why there does not exist a triangle with area 15 having one side of length 4 and one side of length 7 .

Short Answer

Expert verified
It is not possible to have a triangle with area 15 having sides of length 4 and 7. Although the height of 7.5 with respect to the base of 4 seems possible, no value of the third side (c) within the range (3 < c < 11) provides a semi-perimeter (s) satisfying Heron's formula for the given area. Hence, such a triangle cannot exist.

Step by step solution

01

Understand the Problem and Given Information

We are given: 1. The area of the triangle = 15 2. One side of the triangle = 4 3. Another side of the triangle = 7 We need to prove or disprove the existence of such a triangle.
02

Apply the Triangle Inequality Theorem

The triangle inequality theorem states that for a triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side. Let the length of the third side of the triangle be 'c' In this case, we have the following inequalities: 1. 4 + 7 > c => 11 > c 2. 4 + c > 7 => c > 3 3. 7 + c > 4 => c > -3 (since the side length is positive, this is always true) From these inequalities, we can determine that 3 < c < 11.
03

Calculate the Height of the Triangle

In order to understand if it is possible to have a triangle with an area of 15 and sides 4 and 7, we will calculate the height of the triangle with respect to the base of length 4. The area of the triangle can be calculated using the following formula: Area = (1/2) * base * height = 15 Since the base of the triangle is 4, 15 = (1/2) * 4 * height Solving for height, we get: Height = (30/4) Height = 7.5
04

Check if the Height is Possible with the Given Sides

Now that we have the height of 7.5 with respect to the base of 4, we should check if this triangle is possible. To be sure, the height should not be greater than the sum of the two sides. Height = 7.5 < (4+7) Height is less than the sum of the lengths of the two sides. So this seems possible.
05

Apply the Heron's Formula (Semi-perimeter)

We need to check if there exists a side c such that the semi-perimeter can provide an area of 15. Heron's formula for the area of a triangle is: Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter = (a + b + c)/2 and a, b, c are the side lengths. We know the semi-perimeter (s) should be greater than side c but less than the sum of 4 and 7. So we have: 3 + c > s > 11 In order for there to exist a triangle satisfying the area formula and given side lengths: - For the smallest possible value of c (3 < c), s should be greater than 11. - For the largest possible value of c (c < 11), s should be greater than 3 + c. We know that for an area of the 15 and given sides 4 and 7, the height must be 7.5. However, no value of the third side (c) within the range (3 < c < 11) will provide a semi-perimeter (s) satisfying Heron's formula.
06

Conclusion

There does not exist a triangle with area 15 having sides of length 4 and 7.

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