The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Using this theorem, we can check which sets of measures will make a triangle a right triangle.
a) 6, 8, 10
Using the Pythagorean theorem:
6^2 + 8^2 = 36 + 64 = 100
10^2 = 100
Since 6^2 + 8^2 = 10^2, this set of measures will make a right triangle.
b) 12, 16, 20
Using the Pythagorean theorem:
12^2 + 16^2 = 144 + 256 = 400
20^2 = 400
Since 12^2 + 16^2 = 20^2, this set of measures will make a right triangle.
c) 5, 10, 15
Using the Pythagorean theorem:
5^2 + 10^2 = 25 + 100 = 125
15^2 = 225
Since 5^2 + 10^2 is not equal to 15^2, this set of measures will not make a right triangle.
d) 10, 24, 26
Using the Pythagorean theorem:
10^2 + 24^2 = 100 + 576 = 676
26^2 = 676
Since 10^2 + 24^2 = 26^2, this set of measures will make a right triangle.
Therefore, the sets of measures that will make a triangle a right triangle are a) 6, 8, 10 and d) 10, 24, 26. The answer is (a) and (d).