Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( c^2 = 6^2 + 8^2 = 36 + 64 = 100 \)

\( c = \sqrt{100} = 10 \)

Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( 13^2 = 12^2 + b^2 \)

\( 169 = 144 + b^2 \)

\( b^2 = 169 - 144 = 25 \)

\( b = \sqrt{25} = 5 \)

Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( 15^2 = 9^2 + 12^2 \)

\( 225 = 81 + 144 \)

\( 225 = 225 \)

Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( c^2 = 9^2 + 12^2 = 81 + 144 = 225 \)

\( c = \sqrt{225} = 15 \)

The distance formula is derived from the Pythagorean Theorem:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

\( d = \sqrt{(4 - 1)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( 17^2 = 8^2 + b^2 \)

\( 289 = 64 + b^2 \)

\( b^2 = 289 - 64 = 225 \)

\( b = \sqrt{225} = 15 \)

For a square, diagonal \( c \) can be found using \( c^2 = a^2 + a^2 = 2a^2 \):

\( c^2 = 2(7^2) = 2(49) = 98 \)

\( c = \sqrt{98} \approx 9.90 \)

Using the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

\( 25^2 = 7^2 + b^2 \)

\( 625 = 49 + b^2 \)

\( b^2 = 625 - 49 = 576 \)

\( b = \sqrt{576} = 24 \)

The distance formula in 3D is derived from the Pythagorean Theorem:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

\( d = \sqrt{(4 - 1)^2 + (6 - 2)^2 + (7 - 3)^2} \)

\( d = \sqrt{3^2 + 4^2 + 4^2} = \sqrt{9 + 16 + 16} = \sqrt{41} \approx 6.40 \)

The area of a right triangle is given by:

\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

\( \text{Area} = \frac{1}{2} \times 9 \times 12 = \frac{1}{2} \times 108 = 54 \)