Use the Pythagorean Theorem Quick Check1 of 51 of 5 ItemsQuestionUse the image to answer the question.A right triangle shows the base with a value of 3, the hypotenuse on the left with a value of 15, and the perpendicular side to the base on the right with a question mark.Apply the Pythagorean Theorem to find the unknown side length in the given triangle. Round your answer to the nearest tenth.(1 point)Responses21621623423415.315.314.7
To find the length of the hypotenuse, 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 two legs.
For this triangle with legs of length 12 cm, we have:
\(a = 12\) cm
\(b = 12\) cm
\(c\) (hypotenuse) is the unknown.
Using the Pythagorean theorem, we get:
\(a^2 + b^2 = c^2\)
\(12^2 + 12^2 = c^2\)
\(144 + 144 = c^2\)
\(288 = c^2\)
To solve for \(c\), we take the square root of both sides:
\(c = \sqrt{288} \approx 16.97\) cm
Therefore, the length of the hypotenuse is approximately 16.97 cm.
To find the height that the ladder will reach, 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 two legs.
In this case, one leg of the triangle is the distance from the base of the ladder to the house, which is 6 ft. The other leg is the height that the ladder will reach, which is the unknown.
Let's use \(a\) to represent the distance from the base of the ladder to the house (6 ft) and \(b\) to represent the height that the ladder will reach.
Using the Pythagorean theorem, we have:
\(a^2 + b^2 = c^2\)
\(6^2 + b^2 = 12^2\)
\(36 + b^2 = 144\)
\(b^2 = 144 - 36\)
\(b^2 = 108\)
To solve for \(b\), we take the square root of both sides:
\(b = \sqrt{108} \approx 10.4\) ft
Therefore, the ladder will reach a height of approximately 10.4 ft.
To find the distance from the base of the pole to where each bracket should be braced in the ground, 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 two legs.
In this case, one leg of the triangle is the height of the point on the pole where the bracket will be attached, which is 4.5 ft. The other leg is the length of the bracket, which is 6.5 ft.
Let's use \(a\) to represent the distance from the base of the pole to where the bracket should be braced in the ground.
Using the Pythagorean theorem, we have:
\(a^2 + 4.5^2 = 6.5^2\)
\(a^2 + 20.25 = 42.25\)
\(a^2 = 42.25 - 20.25\)
\(a^2 = 22\)
To solve for \(a\), we take the square root of both sides:
\(a = \sqrt{22} \approx 4.7\) ft
Therefore, each bracket should be braced in the ground approximately 4.7 ft from the base of the pole.
To apply the Pythagorean theorem, we can use the equation:
\(a^2 + b^2 = c^2\)
where \(a\) and \(b\) are the lengths of the two legs of the right triangle, and \(c\) is the length of the hypotenuse.
In this case, the base has a length of 3, the hypotenuse has a length of 15, and the unknown side (perpendicular to the base) is represented by \(b\).
So we have:
\(3^2 + b^2 = 15^2\)
\(9 + b^2 = 225\)
Now, let's solve for \(b\):
\(b^2 = 225 - 9\)
\(b^2 = 216\)
Taking the square root of both sides:
\(b = \sqrt{216} \approx 14.7\)
Therefore, the unknown side length is approximately 14.7.
To find the length of the cable, 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 two legs.
In this case, one leg of the triangle is the distance from the base of the pole to the anchor point on the ground, which is 10 ft. The other leg is the height of the pole above the anchor point, which is 20 ft.
Let's use \(a\) to represent the distance from the base of the pole to the anchor point (10 ft) and \(b\) to represent the height of the pole above the anchor point.
Using the Pythagorean theorem, we have:
\(a^2 + b^2 = c^2\)
\(10^2 + 20^2 = c^2\)
\(100 + 400 = c^2\)
\(500 = c^2\)
To solve for \(c\), we take the square root of both sides:
\(c = \sqrt{500} \approx 22.4\) ft
Therefore, the length of the cable is approximately 22.4 ft.