Question

name (first and last): philip vusirar period: 5 date: 10/6/28 special right triangles version b - criterion b: pattern investigation. using pythagoreans theorem and exact answers, determine the value of x and sometimes y in each triangle. 1) 6² + 6² = x² 2) 45° 45° 21 21 3) 3√49 7 7√2 45° x 4) 19² + 19² = x² 19 19 45° for problems 5 and 6, solve for x and y. leave your answers in exact form. 5) 14 y x 6) 4√2 c = 4√2 l = a√2 7) what do the first 6 triangles all have in common? what patterns do you notice?

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Solve problem 1

Given \(a = 6\) and \(b = 6\), then \(c=x\). So \(6^{2}+6^{2}=x^{2}\), \(36 + 36=x^{2}\), \(x^{2}=72\), \(x=\sqrt{72}=6\sqrt{2}\).

Step3: Solve problem 2

Given \(a = 21\) and \(b = 21\), then \(c=x\). So \(21^{2}+21^{2}=x^{2}\), \(441+441=x^{2}\), \(x^{2}=882\), \(x=\sqrt{882}=21\sqrt{2}\).

Step4: Solve problem 3

Given \(a = 7\) and \(c = 7\sqrt{2}\), then \(a^{2}+b^{2}=c^{2}\), \(7^{2}+x^{2}=(7\sqrt{2})^{2}\), \(49+x^{2}=98\), \(x^{2}=49\), \(x = 7\).

Step5: Solve problem 4

Given \(a = 19\) and \(b = 19\), then \(c=x\). So \(19^{2}+19^{2}=x^{2}\), \(361+361=x^{2}\), \(x^{2}=722\), \(x=\sqrt{722}=19\sqrt{2}\).

Step6: Solve problem 5

Given \(a = 14\), \(b=x\), \(c = y\). If it is a \(45 - 45-90\) triangle, \(x = 14\) and \(y=\sqrt{14^{2}+14^{2}}=\sqrt{196 + 196}=\sqrt{392}=14\sqrt{2}\).

Step7: Solve problem 6

Given \(c = 4\sqrt{2}\) in a \(45 - 45-90\) triangle, if the legs are \(a=x\) and \(b = y\), then \(a=b\) and \(c^{2}=a^{2}+b^{2}=2a^{2}\). So \((4\sqrt{2})^{2}=2a^{2}\), \(32 = 2a^{2}\), \(a^{2}=16\), \(a=x = 4\), \(y = 4\).

Step8: Analyze commonalities

The first 6 triangles are all right - triangles. Triangles 1, 2, 4 are isosceles right - triangles (\(45 - 45-90\) triangles) where if the leg length is \(l\), the hypotenuse length is \(l\sqrt{2}\). In problem 3, it is also a right - triangle with a known leg and hypotenuse relationship, and problem 5 and 6 are right - triangles with side - length relationships based on the Pythagorean theorem and properties of special right - triangles.

Answer:

1. \(x = 6\sqrt{2}\)
2. \(x=21\sqrt{2}\)
3. \(x = 7\)
4. \(x=19\sqrt{2}\)
5. \(x = 14\), \(y = 14\sqrt{2}\)
6. \(x = 4\), \(y = 4\)
7. They are all right - triangles. Triangles 1, 2, 4 are \(45 - 45-90\) triangles where hypotenuse \(=\) leg \(\times\sqrt{2}\).