From his house, Gabe could drive due north to get to his parents’ house or he could drive due east to get to his grandparents’ house. It is 8.9 miles from Gabe’s house to his parents’ house and a straight-line distance of 9.5 miles from his parents’ house to his grandparents’ house. How far is Gabe from his grandparents’ house? If necessary, round to the nearest tenth.
miles
To find the distance from Gabe’s house to his grandparents’ house, we can use the Pythagorean theorem, since Gabe’s house, his parents’ house, and his grandparents’ house can be represented as a right triangle.
1. Gabe’s house to his parents’ house is 8.9 miles (north).
2. The distance from his parents’ house to his grandparents’ house is 9.5 miles (east).
We need to find the hypotenuse (Gabe’s house to his grandparents’ house).
Using the Pythagorean theorem:
[ c^2 = a^2 + b^2 ]
where:
– ( c ) is the distance from Gabe’s house to his grandparents’ house,
– ( a ) is 8.9 miles,
– ( b ) is 9.5 miles.
Calculating:
[ c^2 = (8.9)^2 + (9.5)^2 ]
[ c^2 = 79.21 + 90.25 ]
[ c^2 = 169.46 ]
Now, take the square root to find ( c ):
[ c = sqrt{169.46} ]
[ c approx 13.0 , text{miles} ]
Thus, Gabe is approximately 13.0 miles from his grandparents’ house.