The Pythagorean Theorem: Grade 8 Math Test Answer Key – PDF Download

Question 1: Hypotenuse (2 points)

\(40^{2}+9^{2}=1600+81=1681\), so \(c=\sqrt{1681}=41\). The hypotenuse is 41 in.

Scoring: 1 point for the correct equation \(40^{2}+9^{2}=c^{2}\), 1 point for 41 in.

Question 2: Missing leg (2 points)

\(10^{2}+b^{2}=26^{2}\), so \(b^{2}=676-100=576\) and \(b=24\). The other leg is 24 ft.

Scoring: 1 point for subtracting the squares correctly (576), 1 point for 24 ft.

Question 3: Rounded hypotenuse (2 points)

\(5^{2}+9^{2}=25+81=106\), so \(c=\sqrt{106}\approx 10.30\). The hypotenuse is about 10.3 in.

Scoring: 1 point for \(c^{2}=106\), 1 point for the correctly rounded answer.

Question 4: Right triangle or not (3 points)

1. \(12^{2}+35^{2}=144+1225=1369=37^{2}\). Right triangle.
2. \(10^{2}+24^{2}=100+576=676\), but \(27^{2}=729\). Not a right triangle.
3. \(16^{2}+30^{2}=256+900=1156=34^{2}\). Right triangle.

Scoring: 1 point per part: the calculation with the longest side as the hypotenuse and the correct conclusion.

Question 5: Area proof (2 points)

1. Big square: \(7^{2}=49\). Triangles: \(4\cdot\frac{1}{2}\cdot 3\cdot 4=24\). Tilted square: \(49-24=\)25 square inches.
2. \(\sqrt{25}=\)5 in.

Scoring: 1 point for each part.

Question 6: Ladder (2 points)

\(x^{2}+15^{2}=17^{2}\), so \(x^{2}=289-225=64\) and \(x=8\). The base is 8 ft from the wall.

Scoring: 1 point for the equation with 17 as the hypotenuse, 1 point for 8 ft.

Question 7: TV screen (3 points)

1. \(40^{2}+h^{2}=50^{2}\), so \(h^{2}=2500-1600=900\) and \(h=\)30 in.
2. \(40\cdot 30=\)1,200 square inches.

Scoring: Part (a): 1 point for the equation, 1 point for 30 in. Part (b): 1 point.

Question 8: Rod in a box (2 points)

Base diagonal: \(x^{2}=24^{2}+10^{2}=576+100=676\), so \(x=26\). Space diagonal: \(d^{2}=26^{2}+6^{2}=676+36=712\), so \(d\approx 26.7\) in.

Since 26.7 is less than 30, the rod does not fit.

Scoring: 1 point for finding the space diagonal (\(d^{2}=712\)), 1 point for comparing with 30 and concluding no.

Question 9: Cutting across the park (2 points)

Diagonal: \(120^{2}+50^{2}=14400+2500=16900\), so \(d=130\) yd. Along two sides: \(120+50=170\) yd.

Difference: \(170-130=\)40 yd shorter.

Scoring: 1 point for the diagonal length 130 yd, 1 point for the difference 40 yd.

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