Use the Pythagorean theorem to determine the side lengths of a right triangle. A crucial tool in physics, building, and geometry, you can enter any two sides to find the third.
The hypotenuse's square (c) in a right triangle is equal to the sum of the squares of the other two sides (a and b):
c⊃2; = a⊃2; + b⊃2;
c = √(a⊃2; + b⊃2;)
a = √(c⊃2; - b⊃2;)
b = √(c⊃2; - a⊃2;)
The Pythagorean Theorem is used in various fields such as:
A right triangle has legs of lengths 6 and 8. What is the length of the hypotenuse?
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( c^2 = 6^2 + 8^2 = 36 + 64 = 100 \)
\( c = \sqrt{100} = 10 \)
Answer: The hypotenuse is 10.A right triangle has a hypotenuse of 13 and one leg of length 12. Find the other leg.
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( 13^2 = 12^2 + b^2 \)
\( 169 = 144 + b^2 \)
\( b^2 = 169 - 144 = 25 \)
\( b = \sqrt{25} = 5 \)
Answer: The other leg is 5.A triangle has sides of lengths 9, 12, and 15. Is it a right triangle?
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( 15^2 = 9^2 + 12^2 \)
\( 225 = 81 + 144 \)
\( 225 = 225 \)
Answer: Yes, it is a right triangle.A rectangle has a length of 9 and a width of 12. Find the length of its diagonal.
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( c^2 = 9^2 + 12^2 = 81 + 144 = 225 \)
\( c = \sqrt{225} = 15 \)
Answer: The diagonal is 15.Find the distance between the points (1, 1) and (4, 5) on a coordinate plane.
Show SolutionThe distance formula is derived from the Pythagorean Theorem:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
\( d = \sqrt{(4 - 1)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Answer: The distance is 5 units.A right triangle has a hypotenuse of 17 and one leg of length 8. Find the height (other leg) of the triangle.
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( 17^2 = 8^2 + b^2 \)
\( 289 = 64 + b^2 \)
\( b^2 = 289 - 64 = 225 \)
\( b = \sqrt{225} = 15 \)
Answer: The other leg is 15.A square has sides of length 7. Find the length of its diagonal.
Show SolutionFor a square, diagonal \( c \) can be found using \( c^2 = a^2 + a^2 = 2a^2 \):
\( c^2 = 2(7^2) = 2(49) = 98 \)
\( c = \sqrt{98} \approx 9.90 \)
Answer: The diagonal is approximately 9.90.A ladder is 25 feet long and is placed against a wall. The base of the ladder is 7 feet away from the wall. How high up the wall does the ladder reach?
Show SolutionUsing the Pythagorean Theorem: \( c^2 = a^2 + b^2 \)
\( 25^2 = 7^2 + b^2 \)
\( 625 = 49 + b^2 \)
\( b^2 = 625 - 49 = 576 \)
\( b = \sqrt{576} = 24 \)
Answer: The ladder reaches 24 feet up the wall.Find the distance between the points (1, 2, 3) and (4, 6, 7) in 3D space.
Show SolutionThe distance formula in 3D is derived from the Pythagorean Theorem:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)
\( d = \sqrt{(4 - 1)^2 + (6 - 2)^2 + (7 - 3)^2} \)
\( d = \sqrt{3^2 + 4^2 + 4^2} = \sqrt{9 + 16 + 16} = \sqrt{41} \approx 6.40 \)
Answer: The distance is approximately 6.40 units.A right triangle has legs of lengths 9 and 12. Find its area.
Show SolutionThe area of a right triangle is given by:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
\( \text{Area} = \frac{1}{2} \times 9 \times 12 = \frac{1}{2} \times 108 = 54 \)
Answer: The area is 54 square units.© 2025 Calculatoric.com All rights reserved
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