Question

4 use the converse of the pythagorean theorem to show that the triangle is a right triangle.
(image of a right triangle with legs 9 and 40, hypotenuse 41)
5 use the converse of the pythagorean theorem to determine if the given side lengths could be used to form a right triangle. explain your reasoning.
( a = 12 ) ( b = 16 ) ( c = 18 )
6 a right triangle is shown. what is missing side length? show your work.
(image of a right triangle with one leg 12, hypotenuse 37, and the other leg labeled ( a ))

Explanation:

Problem 4

Step1: Recall the converse of the Pythagorean theorem.

The converse of the Pythagorean theorem states that if \(a^2 + b^2 = c^2\) for a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), then the triangle is a right triangle. Here, the sides are \(a = 40\), \(b = 9\), and \(c = 41\) (since 41 is the longest side).

Step2: Calculate \(a^2 + b^2\) and \(c^2\).

First, calculate \(a^2\): \(40^2 = 1600\).
Then, calculate \(b^2\): \(9^2 = 81\).
Add them together: \(1600 + 81 = 1681\).
Now, calculate \(c^2\): \(41^2 = 1681\).

Step3: Compare the two results.

Since \(a^2 + b^2 = 1681\) and \(c^2 = 1681\), we have \(a^2 + b^2 = c^2\). By the converse of the Pythagorean theorem, the triangle is a right triangle.

Step1: Identify the longest side.

Among \(a = 12\), \(b = 16\), and \(c = 18\), the longest side is \(c = 18\). So we check if \(a^2 + b^2 = c^2\) (converse of Pythagorean theorem).

Step2: Calculate \(a^2\), \(b^2\), and \(c^2\).

Calculate \(a^2\): \(12^2 = 144\).
Calculate \(b^2\): \(16^2 = 256\).
Calculate \(c^2\): \(18^2 = 324\).

Step3: Calculate \(a^2 + b^2\) and compare to \(c^2\).

Add \(a^2\) and \(b^2\): \(144 + 256 = 400\).
Now, compare to \(c^2 = 324\). Since \(400
eq 324\), \(a^2 + b^2
eq c^2\).

Step4: Conclusion.

By the converse of the Pythagorean theorem, since \(a^2 + b^2
eq c^2\), the side lengths \(12\), \(16\), and \(18\) do not form a right triangle.

Step1: Recall the Pythagorean theorem.

For a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse (longest side), and \(a\) and \(b\) are the legs. Here, one leg is \(12\), the hypotenuse \(c = 37\), and we need to find the other leg \(a\). So we rearrange the formula: \(a^2 = c^2 - b^2\) (assuming \(b = 12\)).

Step2: Substitute the known values.

Substitute \(c = 37\) and \(b = 12\) into the formula: \(a^2 = 37^2 - 12^2\).

Step3: Calculate \(37^2\) and \(12^2\).

Calculate \(37^2\): \(37 \times 37 = 1369\).
Calculate \(12^2\): \(12 \times 12 = 144\).

Step4: Subtract to find \(a^2\).

Subtract: \(a^2 = 1369 - 144 = 1225\).

Step5: Take the square root to find \(a\).

Take the square root of \(1225\): \(a = \sqrt{1225} = 35\).

Answer:

The triangle is a right triangle because \(40^2 + 9^2 = 41^2\) (i.e., \(1600 + 81 = 1681\)).

Problem 5