Question

a window is marked with a grid that acts as a coordinate plane. a fly and a mosquito land on the window. complete the parts below to find the distance between the two insects. (a) the fly is at (-1, -6) and the mosquito is at (1, -1). draw a right triangle so that the hypotenuse is the segment between these two points. (b) let x be the length of the hypotenuse. use the triangle you drew and the pythagorean theorem to fill in the blanks. \\(\square^2 + \square^2 = x^2\\) (c) find x, the length of the hypotenuse. give an exact answer (not a decimal approximation). \\(x = \square\\) (d) each unit on the grid has a length of 4 centimeters. use your answer from part (c) to find the distance between the two insects. round your answer to the nearest tenth.

Explanation:

Step1: Determine horizontal and vertical distances

The fly is at \((-1, -6)\) and the mosquito at \((1, -1)\). Horizontal distance: \(|1 - (-1)| = 2\). Vertical distance: \(|-1 - (-6)| = 5\).

Step2: Apply Pythagorean Theorem

For a right triangle, \(a^2 + b^2 = c^2\), where \(a = 2\), \(b = 5\), \(c = x\). So \(2^2 + 5^2 = x^2\).

Step3: Calculate \(x^2\)

\(4 + 25 = x^2\) → \(x^2 = 29\).

Step4: Find \(x\) (exact)

\(x = \sqrt{29}\) (exact answer).

Step5: Find decimal approximation (part d)

\(\sqrt{29} \approx 5.4\) (nearest tenth). For part (d) with grid unit = 4 cm: \(5.4 \times 4 = 21.6\) cm (but focusing on part (c) exact first).

Answer:

(for part c):
\(\sqrt{29}\)

(For part d, the answer would be approximately \(21.6\) cm, but as per part c, exact is \(\sqrt{29}\))