Determine the length of the missing side of each right triangle. Show your work.
1)a=15; b=?; c=20
Answer is b= 5√7
2)a=9; b=12; c=?
Answer is c= 15
**How do I show my work to come up with each of these two answers?
THANK YOU!!
1. a^2 + b^2 = c^2.
15^2 + b^2 = 20^2,
225 + b^2 = 400.
b^2 = 400-225 = 175,
b = sqrt(25*7) = 5*sqrt7.
2. 9^2 + 12^2 = c^2.
c^2 = 81 + 144 = 225,
c = sqrt(225) = 15.
1)
b = √ ( c ^ 2 - a ^ 2 )
b = √ ( 20 ^ 2 - 15 ^ 2 )
b = √ ( 400 - 225 )
b = √ ( 175 )
b = √ ( 7 * 25 )
b = √ 7 * √ ( 25 )
b = √ 7 * 5
2)
c = √ ( a ^ 2 + b ^ 2 )
c = √ ( 9 ^ 2 + 12 ^ 2 )
c = √ ( 81 + 144 )
c = √ ( 225 )
c = 25
To determine the length of the missing side of each right triangle, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
1) For the first right triangle with side lengths a = 15, b = ?, and c = 20, we can substitute the values into the Pythagorean theorem:
c^2 = a^2 + b^2
20^2 = 15^2 + b^2
400 = 225 + b^2
Now, subtract 225 from both sides:
400 - 225 = b^2
175 = b^2
To solve for b, take the square root of both sides:
√175 = √b^2
b ≈ √175
b ≈ 13.22875 (approximately)
However, if we simplify the square root of 175, we get:
√175 = √25 * √7
√175 = 5 * √7
Therefore, the answer is b ≈ 5√7.
2) For the second right triangle with side lengths a = 9, b = 12, and c = ?, we can again use the Pythagorean theorem:
c^2 = a^2 + b^2
c^2 = 9^2 + 12^2
c^2 = 81 + 144
c^2 = 225
To solve for c, take the square root of both sides:
√c^2 = √225
c = 15
Therefore, the answer is c = 15.
By following these steps, you can show your work to determine the length of the missing side in each right triangle.
To determine the length of the missing side of each right triangle, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
For the first triangle:
a = 15, b = ?, c = 20
To find the missing side length, b, we can use the Pythagorean theorem:
b^2 = c^2 - a^2
b^2 = 20^2 - 15^2
b^2 = 400 - 225
b^2 = 175
To solve for b, we can take the square root of both sides:
b = √175
Simplifying the square root further:
b = √(25 * 7)
b = 5√7
So the length of the missing side, b, is 5√7.
For the second triangle:
a = 9, b = 12, c = ?
To find the missing side length, c, we again use the Pythagorean theorem:
c^2 = a^2 + b^2
c^2 = 9^2 + 12^2
c^2 = 81 + 144
c^2 = 225
Taking the square root of both sides:
c = √225
c = 15
Therefore, the length of the missing side, c, is 15.