The hypotenuse (side C) of a triangle is 13 inches long. Which of the following pairs of measurements could be correct for the lengths of the two sides of the triangle? (Note:A^2 + B^2 = C^2)
Isn't your school subject geometry?
remember the 5-12-13 right triangle.
Of course, there are many other non-integer solutions.
To determine which pairs of measurements could be correct for the lengths of the two sides of the triangle, we can use the Pythagorean theorem, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse.
Let's consider each option:
Option 1: A = 5 inches, B = 12 inches
Applying the Pythagorean theorem: 5^2 + 12^2 = 25 + 144 = 169
Square root of 169 is 13, which matches the length of the hypotenuse (C). Therefore, option 1 is a correct pair of measurements.
Option 2: A = 8 inches, B = 9 inches
Applying the Pythagorean theorem: 8^2 + 9^2 = 64 + 81 = 145
The square root of 145 is not an integer, and it does not match the length of the hypotenuse (C). Therefore, option 2 is not a correct pair of measurements.
Option 3: A = 7 inches, B = 10 inches
Applying the Pythagorean theorem: 7^2 + 10^2 = 49 + 100 = 149
The square root of 149 is not an integer, and it does not match the length of the hypotenuse (C). Therefore, option 3 is not a correct pair of measurements.
Option 4: A = 6 inches, B = 11 inches
Applying the Pythagorean theorem: 6^2 + 11^2 = 36 + 121 = 157
The square root of 157 is not an integer, and it does not match the length of the hypotenuse (C). Therefore, option 4 is not a correct pair of measurements.
Based on the analysis, only option 1, A = 5 inches and B = 12 inches, is a correct pair of measurements for the lengths of the two sides of the triangle.
To determine which pairs of measurements could be correct for the lengths of the two sides of the triangle, we can use the Pythagorean theorem. According to the theorem, 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).
In this case, the hypotenuse (C) has a length of 13 inches. So, we can use the Pythagorean theorem to find the valid pairs of side lengths.
We need to check which pairs of positive integers (A and B) satisfy the equation A^2 + B^2 = 13^2.
Starting by taking A = 1, we find:
1^2 + B^2 = 13^2
1 + B^2 = 169
B^2 = 168
Since there are no perfect squares less than 168, we move on to A = 2:
2^2 + B^2 = 13^2
4 + B^2 = 169
B^2 = 165
If we continue this process for A = 3, 4, 5, and so on, we eventually find that the pairs of measurements that could be correct for the lengths of the two sides are:
A = 5, B = 12
A = 12, B = 5
These two pairs satisfy the equation A^2 + B^2 = 13^2.
Therefore, the valid pairs of measurements for the lengths of the two sides of the triangle are (5, 12) and (12, 5).