8th Grade The Pythagorean Theorem Test Solutions and Scoring (PDF)

8th Grade math test solutions: The Pythagorean Theorem – Maths PDF

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This pythagorean theorem test answers for 8th Grade covers the pythagorean theorem (Common Core 8.G.B.6, 8.G.B.7), with a free PDF to print.

Use this answer key to score the the Pythagorean theorem test out of 20 points. Each solution shows the full work and a scoring guide, so partial credit can be given fairly when a student shows correct steps but makes an arithmetic slip.

Grade 8 · Common Core State Standards: 8.G.B.6, 8.G.B.7

Pythagorean theorem test answers: worked solutions

Question 1 – Find the hypotenuse [2 pts]

A right triangle.

\(81 + 144 = 225\), so \(c = 15\).

Scoring: 1 point for the equation. 1 point for 15 cm.

Question 2 – Find a leg [2 pts]

\(a^{2} = 17^{2} – 15^{2} = 289 – 225 = 64\), so \(a = 8\) inches.

Scoring: 1 point for the equation. 1 point for 8 in.

Question 3 – Right triangle? [3 pts]

\(64 + 225 = 289 = 17^{2}\): yes.
\(36 + 81 = 117\), but \(144\): no.

Scoring: Part a: 1 point for the sums, 1 point for yes. Part b: 1 point.

Question 4 – Ladder [3 pts]

The ladder reaches 8 ft.

\(6^{2} + h^{2} = 10^{2}\), so \(h^{2} = 64\) and \(h = 8\).

Scoring: 1 point for the setup, 1 point for \(h^{2} = 64\), 1 point for 8 ft.

Question 5 – Estimate a length [3 pts]

Diagonal: \(\sqrt{196 + 36} = \sqrt{232} \approx 15.2\) feet.

Fencing: \(2(14 + 6) = 40\) feet.

Scoring: 1 point for the equation, 1 point for 15.2 ft, 1 point for 40 ft.

Question 6 – 3D [3 pts]

Floor diagonal: \(\sqrt{64 + 25} = \sqrt{89} \approx 9.4\).

Space diagonal: \(\sqrt{64 + 36 + 25} = \sqrt{125} \approx 11.2\).

Scoring: 1 point for each diagonal setup, 1 point for both values.

Question 7 – Shortcut [2 pts]

Diagonal: \(\sqrt{64 + 225} = 17\). Saved: \(23 – 17 = 6\) yards.

Scoring: 1 point for 17 yards. 1 point for 6 yards.

Question 8 – Explain [2 pts]

The hypotenuse is opposite the right angle. Since \(c^{2} = a^{2} + b^{2}\) is larger than \(a^{2}\) or \(b^{2}\), \(c\) is larger than each leg.

Scoring: 1 point for opposite the right angle. 1 point for the reason.

Total score

Total

20 points

Key facts behind this pythagorean theorem test answers

Key facts

\(a^{2} + b^{2} = c^{2}\), where \(c\) is the hypotenuse.
To find a leg: \(a^{2} = c^{2} – b^{2}\).
Converse: if \(a^{2} + b^{2} = c^{2}\), the triangle has a right angle.
Common triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.
The theorem works in 3D: the space diagonal satisfies \(d^{2} = a^{2} + b^{2} + c^{2}\).

How to use this pythagorean theorem test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Grade 8 chapters: Grade 8 math.

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