The Pythagorean Theorem: math test solutions, Grade 8 – download the PDF

1 Missing sides / 4 pts

1. \(c^2 = 225 + 400 = 625\), so \(c = 25\) cm. (1.5 pts)
2. \(a^2 = 37^2 - 12^2 = 1369 - 144 = 1225\), so \(a = 35\) m. (1.5 pts)
3. \(c^2 = 16 + 81 = 97\), so \(c = \sqrt{97} \approx 9.85\) in. (1 pt)

2 The converse / 3 pts

1. Longest side 34. \(16^2 + 30^2 = 256 + 900 = 1156 = 34^2\). It is a right triangle. (1.5 pts)
2. Longest side 17. \(11^2 + 13^2 = 121 + 169 = 290\), but \(17^2 = 289\). The values differ, so it is not a right triangle. (1.5 pts)

3 Points in the plane / 4 pts

1. The legs are \(10 - (-2) = 12\) and \(8 - (-1) = 9\). \(PQ = \sqrt{144 + 81} = \sqrt{225} = 15\). (2 pts)
2. \(PR\) is horizontal (same \(y\)) and \(RQ\) is vertical (same \(x\)), so the angle at \(R\) is \(90^\circ\) (1 pt). \(PR = 12\), \(RQ = 9\), \(PQ = 15\), so the perimeter is \(12 + 9 + 15 = 36\) units. (1 pt)

4 A ladder and a window / 4 pts

1. \(d^2 = 25^2 - 24^2 = 625 - 576 = 49\), so \(d = 7\) ft. (2 pts)
2. \(h^2 = 25^2 - 15^2 = 625 - 225 = 400\), so \(h = 20\) ft. The ladder reaches 20 ft. (2 pts)

5 A storage crate / 3 pts

1. \(\sqrt{81 + 144} = \sqrt{225} = 15\) in. (1 pt)
2. \(d = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25\) in. (2 pts)

6 True or false? / 2 pts

The longest side is 2 cm. \(1.2^2 + 1.6^2 = 1.44 + 2.56 = 4\) (1 pt) and \(2^2 = 4\). The statement is true by the converse of the Pythagorean theorem. (1 pt)