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A utility crew is securing a vertical pole of height \(24\text{ feet}\) using a taut wire cable attached from the top of the pole to an anchor on level ground.

Part A: If the ground anchor is placed \(10\text{ feet}\) from the base of the pole, what is the length, in feet, of the wire cable?

Part B: County safety regulations require the cable length to be at least \(26\text{ feet}\). If the anchor is moved to \(18\text{ feet}\) from the base, does the new cable length meet the safety requirement? Explain with calculations.

Work Space & Final Answers:

Pythagorean Theorem Review • Grade 10 Page 2 of 2 • MCAS Mathematics

Thought Process

Diagonal ramp length is hypotenuse (\(c = 13\)). Horizontal base is leg (\(a = 12\)). Vertical rise is leg \(b\).

\(12^2 + b^2 = 13^2\)
\(144 + b^2 = 169 \implies b^2 = 25\)
\(b = \sqrt{25}\)
Height = 5 feet

Part 4: MCAS Exam Practice • Question 7 Standard 8.G.B.7 / G-GPE.B.7

7. Straight-line distance between \(J(-3, 2)\) and \(K(5, -4)\) Correct: Choice B (10 units)

MCAS Solution Strategy

Plotting points on a coordinate grid forms a right triangle with legs along the horizontal and vertical grid lines:

• Horizontal leg: \(|5 - (-3)| = |5 + 3| = 8\text{ units}\)
• Vertical leg: \(|-4 - 2| = |-6| = 6\text{ units}\)

\(d^2 = (\Delta x)^2 + (\Delta y)^2\)
\(d^2 = 8^2 + 6^2 = 64 + 36 = 100\)
\(d = \sqrt{100} = \mathbf{10\text{ units}}\)

Distractor Analysis: Choice A (8) misses vertical change; Choice C (14) adds legs \(8+6\); Choice D (\(\sqrt{28}\)) subtracts \(64-36\).

Pythagorean Theorem Review • Solutions Page 2 of 3

Answer Key • MCAS Solutions 8 & 9

Open Response Items & Scoring Guidance

Page 3 of 3

8. Isosceles Trapezoid Lawn: Top \(14\text{ m}\), Bottom \(26\text{ m}\), Height \(8\text{ m}\) Slanted Side = 10 meters

Deconstruction & Logic

Drop perpendicular heights from the endpoints of the \(14\text{ m}\) top base to the \(26\text{ m}\) bottom base.

The bottom base is divided into a central \(14\text{ m}\) segment and two congruent outer segments: \(\frac{26 - 14}{2} = \frac{12}{2} = 6\text{ m}\) each.

Each right triangle has legs:
\(a = 6\text{ m}\), \(b = 8\text{ m}\)
\(s^2 = 6^2 + 8^2 = 36 + 64 = 100\)
\(s = \sqrt{100} = \mathbf{10\text{ meters}}\)
Each slanted side measures 10 meters.

9. Utility Pole Support Cable (Height = \(24\text{ ft}\)) Part A: 26 ft • Part B: 30 ft (Meets Code)

Part A Solution

Pole leg \(b = 24\text{ ft}\), ground leg \(a = 10\text{ ft}\)
\(c^2 = 10^2 + 24^2 = 100 + 576 = 676\)
\(c = \sqrt{676} = \mathbf{26\text{ feet}}\)

Scale of (5, 12, 13) triple by factor of 2.

Part B Solution & Justification

Pole leg \(b = 24\text{ ft}\), ground leg \(a = 18\text{ ft}\)
\(c^2 = 18^2 + 24^2 = 324 + 576 = 900\)
\(c = \sqrt{900} = \mathbf{30\text{ feet}}\)

Conclusion: Yes, it meets the requirement because \(30\text{ ft} \ge 26\text{ ft}\) (exceeds minimum by \(4\text{ ft}\)).

MCAS Scoring Guidance & Common Misconceptions

Grading Rubric (Open Response):

• Full Credit (2 pts): Correct equation, accurate arithmetic, and correct units.
• Partial Credit (1 pt): Set up correctly but made arithmetic/square root error.

Frequent Student Traps:

• In Q8, using \(26 - 14 = 12\) as the leg instead of dividing by 2 (\(6\text{ m}\)).
• In Q9 Part B, calculating hypotenuse correctly but failing to write a concluding comparison statement.

Pythagorean Theorem Review • Solutions Page 3 of 3