Question

a rectangle has a width of 9 units and a length of 40 units. what is the length of a diagonal?

31 units
39 units
41 units
49 units

Explanation:

Identify the geometric properties

Using the Properties of Rectangles knowledge point

• A rectangle has four right angles (\(90^\circ\)).
• The diagonal splits the rectangle into two congruent right triangles.
• The width and length of the rectangle form the perpendicular legs of these right triangles, while the diagonal forms the hypotenuse.

Apply the Pythagorean theorem

Let the width be \(w = 9\) and the length be \(l = 40\). Let the diagonal be \(d\).

$$ d^2 = w^2 + l^2 $$

$$ d^2 = 9^2 + 40^2 $$

$$ d^2 = 81 + 1600 = 1681 $$

$$ d = \sqrt{1681} = 41 $$

Analyze the incorrect options

• The value \(31\) comes from subtracting the width from the length: \(40 - 9 = 31\).
• The value \(49\) comes from adding the width and the length: \(40 + 9 = 49\).
• The value \(39\) is a common distractor that might arise from miscalculating the square root of a value near \(1600\) (since \(40^2 = 1600\), a value less than \(40\) like \(39\) cannot be the hypotenuse, which must be longer than both legs).

Answer:

• (A) 31 units
• (B) 39 units
• (C) 41 units (Correct answer)
• (D) 49 units