Answer :
Sure! To find the straight-line distance from your starting point to your final point after driving east and then north, we can use the Pythagorean theorem. Here’s a step-by-step solution:
### Step 1: Understand the Problem
You start driving east for 12 miles and then turn left to drive north for 10 miles. We are looking for the straight-line distance from your starting point to the final point. This forms a right-angled triangle where:
- The distance east (12 miles) is one leg of the triangle.
- The distance north (10 miles) is the other leg of the triangle.
### Step 2: Recall the Pythagorean Theorem
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, which represents our straight-line distance) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is given by:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
### Step 3: Assign Values
- [tex]\( a = 12 \)[/tex] miles (distance driven east)
- [tex]\( b = 10 \)[/tex] miles (distance driven north)
- [tex]\( c \)[/tex] is the straight-line distance (the hypotenuse) we need to find.
### Step 4: Apply the Pythagorean Theorem
Plug the values into the theorem:
[tex]\[ c^2 = 12^2 + 10^2 \][/tex]
[tex]\[ c^2 = 144 + 100 \][/tex]
[tex]\[ c^2 = 244 \][/tex]
[tex]\[ c = \sqrt{244} \][/tex]
### Step 5: Calculate the Square Root
To find [tex]\( c \)[/tex], take the square root of 244:
[tex]\[ c = \sqrt{244} \approx 15.6205 \][/tex]
### Step 6: Round to the Nearest Tenth
Round [tex]\( 15.6205 \)[/tex] to the nearest tenth:
[tex]\[ c \approx 15.6 \][/tex]
### Conclusion
Your straight-line distance from your starting point, after driving 12 miles east and then 10 miles north, is approximately 15.6 miles.
So, the answer is:
[tex]\[ \text{15.6 miles} \][/tex]
### Step 1: Understand the Problem
You start driving east for 12 miles and then turn left to drive north for 10 miles. We are looking for the straight-line distance from your starting point to the final point. This forms a right-angled triangle where:
- The distance east (12 miles) is one leg of the triangle.
- The distance north (10 miles) is the other leg of the triangle.
### Step 2: Recall the Pythagorean Theorem
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, which represents our straight-line distance) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is given by:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
### Step 3: Assign Values
- [tex]\( a = 12 \)[/tex] miles (distance driven east)
- [tex]\( b = 10 \)[/tex] miles (distance driven north)
- [tex]\( c \)[/tex] is the straight-line distance (the hypotenuse) we need to find.
### Step 4: Apply the Pythagorean Theorem
Plug the values into the theorem:
[tex]\[ c^2 = 12^2 + 10^2 \][/tex]
[tex]\[ c^2 = 144 + 100 \][/tex]
[tex]\[ c^2 = 244 \][/tex]
[tex]\[ c = \sqrt{244} \][/tex]
### Step 5: Calculate the Square Root
To find [tex]\( c \)[/tex], take the square root of 244:
[tex]\[ c = \sqrt{244} \approx 15.6205 \][/tex]
### Step 6: Round to the Nearest Tenth
Round [tex]\( 15.6205 \)[/tex] to the nearest tenth:
[tex]\[ c \approx 15.6 \][/tex]
### Conclusion
Your straight-line distance from your starting point, after driving 12 miles east and then 10 miles north, is approximately 15.6 miles.
So, the answer is:
[tex]\[ \text{15.6 miles} \][/tex]
