Question

what do you get when you cross a computer with a lifeguard? write the letter of each answer in the box containing the exercise number. complete the sentence. 1. in a(n) _ triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. 2. a(n) _ triple is a set of three positive integers, a, b, and c, that satisfy the equation c² = a² + b². 3. if the square of the length of the _ side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. 4. for any △abc, where c is the length of the longest side, if c² < a² + b², then △abc is _. 5. for any △abc, where c is the ___ of the longest side, if c² > a² + b², then △abc is obtuse. find the value of x. 6. a = 20, b = 99; find x. 7. a = 60, b = 91; find x. given the side lengths, determine whether the triangle is (1) a 8. 20, 21, and 29 9. 15, 19, and 24

Explanation:

Step1: Recall Pythagorean - related concepts

1. In a right - triangle, by the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. So the answer for 1 is right.

Step2: Define Pythagorean triple

A Pythagorean triple is a set of three positive integers \(a\), \(b\), and \(c\) that satisfy the equation \(c^{2}=a^{2}+b^{2}\). So the answer for 2 is Pythagorean.

Step3: Apply Pythagorean theorem converse

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right - triangle. So the answer for 3 is longest.

Step4: Use triangle inequality for acute triangles

For any \(\triangle ABC\), where \(c\) is the length of the longest side, if \(c^{2}

Step5: Use triangle inequality for obtuse triangles

For any \(\triangle ABC\), where \(c\) is the length of the longest side, if \(c^{2}>a^{2}+b^{2}\), then \(\triangle ABC\) is obtuse. So the answer for 5 is length.

Step6: Apply Pythagorean theorem for 6

Given \(a = 20\), \(b = 99\) in a right - triangle, by the Pythagorean theorem \(x=\sqrt{a^{2}+b^{2}}=\sqrt{20^{2}+99^{2}}=\sqrt{400 + 9801}=\sqrt{10201}=101\).

Step7: Apply Pythagorean theorem for 7

Given \(a = 60\), \(b = 91\) in a right - triangle, by the Pythagorean theorem \(x=\sqrt{a^{2}+b^{2}}=\sqrt{60^{2}+91^{2}}=\sqrt{3600+8281}=\sqrt{11881}=109\).

Step8: Check triangle type for 8

For side lengths \(a = 20\), \(b = 21\), \(c = 29\), \(a^{2}+b^{2}=20^{2}+21^{2}=400 + 441=841\) and \(c^{2}=29^{2}=841\). Since \(a^{2}+b^{2}=c^{2}\), it is a right - triangle.

Step9: Check triangle type for 9

For side lengths \(a = 15\), \(b = 19\), \(c = 24\), \(a^{2}+b^{2}=15^{2}+19^{2}=225+361 = 586\) and \(c^{2}=24^{2}=576\). Since \(a^{2}+b^{2}>c^{2}\), it is an acute - triangle.

Answer:

1. right
2. Pythagorean
3. longest
4. acute
5. length
6. 101
7. 109
8. right - triangle
9. acute - triangle