Every day, Gabrielle bicycles $3.6$ miles due east from her house to school. After school, she bicycles due north from school to her friend Quinn's house, and then she bicycles home on a bike path that goes straight from Quinn's house to her own house. If the distance along the path from Quinn's house to Gabrielle's house is $8.9$ miles, how far does Gabrielle bicycle each day? If necessary, round your answer to the nearest tenth. $\newline$ ____ miles

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Algebra 2

Pythagorean Theorem and its converse

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Q. Every day, Gabrielle bicycles

3.6

miles due east from her house to school. After school, she bicycles due north from school to her friend Quinn's house, and then she bicycles home on a bike path that goes straight from Quinn's house to her own house. If the distance along the path from Quinn's house to Gabrielle's house is

8.9

miles, how far does Gabrielle bicycle each day? If necessary, round your answer to the nearest tenth.

\newline

____ miles

Identify Distances and Shape: Identify the distances Gabrielle travels each day and the shape they form. Gabrielle travels $3.6$ miles east, then north to Quinn's house, and finally $8.9$ miles directly home. This forms a right triangle with the direct path home as the hypotenuse.
Calculate Distance to Quinn's House: Calculate the distance from school to Quinn's house using the Pythagorean theorem. Let $x$ be the distance from school to Quinn's house. We know the hypotenuse is $8.9$ miles and one leg is $3.6$ miles. The equation is: $\newline$ $3.6^2 + x^2 = 8.9^2$ .
Solve for x: Solve for x: $\newline$ $3.6^2 + x^2 = 8.9^2$ $\newline$ $12.96 + x^2 = 79.21$ $\newline$ $x^2 = 79.21 - 12.96$ $\newline$ $x^2 = 66.25$ $\newline$ $x = \sqrt{66.25}$ $\newline$ $x = 8.1$ miles.
Add Total Distance: Add up the total distance Gabrielle bicycles: $3.6$ miles to school, $8.1$ miles to Quinn's house, and $8.9$ miles home. Total distance = $3.6 + 8.1 + 8.9 = 20.6$ miles.