The hypotenuse of a triangle is 13 inches long. Which of the follwoing pairsof measurements could be correct for the lengths of the other two sides of triangle: a:2.5 and 4 inches, b:2.5 and 6 inches,c:5 and 8 inches,d:5 and 12 inches

a^2 + b^2 = c^2

2.5^2 + 4^2 = 13^2

6.25 + 16 = 169 >> Nope!

Try the others the same way.

The hypotenuse is greater than either of the other 2 sides but less than the sum. Therefore the answer is d.

c^2 = 5^2 + 12^2 = 25 + 144 = 169
c = 13.

To determine which sets of measurements could be correct for the lengths of the other two sides of the triangle, we can use the Pythagorean theorem.

The Pythagorean theorem 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 lengths of the other two sides (a and b).

We are given that the hypotenuse (c) is 13 inches long.

Let's check each set of measurements:

a) a = 2.5 inches and b = 4 inches
Using the Pythagorean theorem: 2.5^2 + 4^2 = 6.25 + 16 = 22.25
√22.25 ≈ 4.72 inches
Since the square root of 22.25 is not equal to 13, this set of measurements is not correct.

b) a = 2.5 inches and b = 6 inches
Using the Pythagorean theorem: 2.5^2 + 6^2 = 6.25 + 36 = 42.25
√42.25 ≈ 6.50 inches
Since the square root of 42.25 is not equal to 13, this set of measurements is not correct.

c) a = 5 inches and b = 8 inches
Using the Pythagorean theorem: 5^2 + 8^2 = 25 + 64 = 89
√89 ≈ 9.43 inches
Since the square root of 89 is not equal to 13, this set of measurements is not correct.

d) a = 5 inches and b = 12 inches
Using the Pythagorean theorem: 5^2 + 12^2 = 25 + 144 = 169
√169 = 13 inches
Since the square root of 169 is equal to 13, this set of measurements is correct.

Therefore, the correct set of measurements for the lengths of the other two sides of the triangle is d) 5 and 12 inches.

To determine which pairs of measurements could be correct for the lengths of the other two sides of the triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's calculate the squares of the lengths of the other two sides of the triangle for each pair of measurements:

a: (2.5)^2 + (4)^2 = 6.25 + 16 = 22.25
b: (2.5)^2 + (6)^2 = 6.25 + 36 = 42.25
c: (5)^2 + (8)^2 = 25 + 64 = 89
d: (5)^2 + (12)^2 = 25 + 144 = 169

Now, let's compare the sum of the squares to the square of the length of the hypotenuse (13^2 = 169):

a: 22.25 is less than 169
b: 42.25 is less than 169
c: 89 is less than 169
d: 169 is equal to 169

From the calculations, we can see that pair d: 5 and 12 inches is the only one where the sum of the squares of the lengths of the other two sides is equal to the square of the hypotenuse. Therefore, the correct pair of measurements for the lengths of the other two sides of the triangle is d: 5 and 12 inches.