What is the relationship of the sides in this triangle based on the Pythagorean Theorem? (1 point)
m^2 + n^2 =
c^2
If a triangle has sides measuring 60, 75, and 45, is it a right triangle?
A. Yes, because 5,625 equals 5,625
B. No, because 9,225 does not equal 2,025
C. No, because 5,625 does not equal 3,600
D. Yes, because 3,600 equals 3,600
C. No, because 5,625 does not equal 3,600
If triangle DEF has sides measuring 50, 40, and 32, is the triangle a right triangle? If it is a right triangle, answer with the number 90. If it is not a right triangle, answer with the number 0.
To determine if the triangle is a right triangle, we can check if the square of the longest side is equal to the sum of the squares of the other two sides.
Sum of squares of shorter sides: 40^2 + 32^2 = 1600 + 1024 = 2624
Square of longest side: 50^2 = 2500
Since 2624 is not equal to 2500, the triangle DEF is not a right triangle. Therefore, the answer is 0.
If a triangle has sides of length 12 and 16, how long is the hypotenuse?
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 other two sides.
So, in this case:
Length of one side = 12
Length of another side = 16
Let's assume the length of the hypotenuse is 'c'.
Using the Pythagorean theorem:
c^2 = 12^2 + 16^2
c^2 = 144 + 256
c^2 = 400
To find the length of the hypotenuse, we need to take the square root of both sides:
c = √400
c = 20
Therefore, the hypotenuse has a length of 20.
If a right triangle has a leg with a length of 12 and the hypotenuse is 16, how long is the other leg? Round to the nearest tenth, if necessary
To find the length of the other leg in a right 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.
So, in this case:
Length of one leg = 12
Length of the hypotenuse = 16
Let's assume the length of the other leg is 'x'.
Using the Pythagorean theorem:
16^2 = 12^2 + x^2
256 = 144 + x^2
x^2 = 256 - 144
x^2 = 112
To find the length of the other leg, we need to take the square root of both sides:
x = √112
x ≈ 10.60 (rounded to the nearest tenth)
Therefore, the other leg has a length of approximately 10.60.
A shortstop is standing in the base path between second and third base when she fields the ball. She is 30 feet from second base and needs to make the throw to first base. Assuming the field is square, how far does she need to throw the ball? Round your answer to the nearest tenth.
Since the field is square, we can use the Pythagorean theorem to find the distance she needs to throw the ball.
We have a right triangle where the distance from second base to first base is the hypotenuse, the distance from the shortstop to second base is one leg, and the distance from the shortstop to first base is the other leg.
Let's assume the distance she needs to throw the ball is 'x'.
Using the Pythagorean theorem:
x^2 = (30)^2 + (90)^2
x^2 = 900 + 8100
x^2 = 9000
To find the distance she needs to throw the ball, we need to take the square root of both sides:
x = √9000
x ≈ 94.868 (rounded to the nearest tenth)
Therefore, she needs to throw the ball approximately 94.9 feet.
To find the height of the pyramid, we can use the Pythagorean theorem.
Let's assume the height of the pyramid is 'h'.
Using the Pythagorean theorem:
h^2 = (180.2)^2 - (230.5/2)^2
h^2 = 32584.04 - 26587.52
h^2 = 5996.52
To find the height of the pyramid, we need to take the square root of both sides:
h = √5996.52
h ≈ 77.4 (rounded to the nearest tenth)
Therefore, the height of the pyramid is approximately 77.4 m.
Keysha is building a coffee table. It's going to be in the shape of a rectangular prism, without the faces. She wants to make the table decorative by adding metal diagonals. The length of the coffee table will be 44 in, the width 24 in, and the height 14 in. Find the length Keysha should make diagonal. Round your answer to the nearest tenth.
To find the length of the diagonal, we can use the Pythagorean theorem.
Let's assume the length of the diagonal is 'd'.
Using the Pythagorean theorem:
d^2 = 44^2 + 24^2 + 14^2
d^2 = 1936 + 576 + 196
d^2 = 2708
To find the length of the diagonal, we need to take the square root of both sides:
d = √2708
d ≈ 52.1 (rounded to the nearest tenth)
Therefore, Keysha should make the diagonal approximately 52.1 inches long.
Apply the Pythagorean Theorem to find the length between the two points (7, 8) and (3, 2). Round your answer to the nearest hundredth.
To find the distance between two points using the Pythagorean theorem, we need to use the difference in x-coordinates and the difference in y-coordinates.
Let's assume the coordinates (7, 8) represent point A and the coordinates (3, 2) represent point B.
The difference in x-coordinates is 7 - 3 = 4.
The difference in y-coordinates is 8 - 2 = 6.
Using the Pythagorean theorem:
Distance^2 = (Difference in x)^2 + (Difference in y)^2
Distance^2 = 4^2 + 6^2
Distance^2 = 16 + 36
Distance^2 = 52
To find the distance between the two points, we need to take the square root of 52:
Distance = √52
Distance ≈ 7.211 (rounded to the nearest hundredth)
Therefore, the distance between the points (7, 8) and (3, 2) is approximately 7.211.
Find the slant height of the square pyramid.
To find the slant height of a square pyramid, we need to know the length of the pyramid's apothem (distance from the center of the base to the midpoint of one of its sides) and the length of one of its triangular faces.
Without these measurements or additional information, it is not possible to determine the slant height of the square pyramid.
Find the slant height of the square pyramid. 15 and 16
To find the slant height of a square pyramid, we can use the Pythagorean theorem.
Let's assume the slant height is 's' and the height of the pyramid is 'h'.
Using the Pythagorean theorem:
s^2 = h^2 + (1/2 * side length)^2
Given the height (h) of 15 and side length (a) of 16, we can substitute these values into the equation:
s^2 = 15^2 + (1/2 * 16)^2
s^2 = 225 + 64
s^2 = 289
To find the slant height, we need to take the square root of 289:
s = √289
s = 17
Therefore, the slant height of the square pyramid is 17.
What is the measure of the exterior angle x?
The other angles are 37 and 42
To find the measure of the exterior angle (x) of a triangle, we can use the fact that the sum of the measures of the interior angles of a triangle is always 180 degrees.
Let's assume the measures of the other two interior angles are 37 and 42 degrees.
The sum of the measures of the interior angles of a triangle is given by:
180 = x + 37 + 42
We can simplify the equation:
180 = x + 79
To solve for x, we subtract 79 from both sides:
x = 180 - 79
x = 101
Therefore, the measure of the exterior angle (x) is 101 degrees.
Explain a proof of the Pythagorean Theorem
The Pythagorean theorem is a fundamental relationship in geometry that 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.
There are several different ways to prove the Pythagorean theorem, but one common and intuitive proof is known as the geometric proof.
The geometric proof starts by drawing a right triangle with sides of lengths a, b, and c, where c is the hypotenuse. Then, we draw three squares, each with a side length corresponding to one of the sides of the triangle.
Now, we can see that the area of the large square (with side c) is equal to the sum of the areas of the other two squares (with sides a and b).
The area of the large square is c^2, and the areas of the smaller squares are a^2 and b^2. We can write this relationship as follows:
c^2 = a^2 + b^2
To visualize this, imagine arranging four copies of the right triangle so that they form a larger square with side length (a + b). Then, the area of this larger square is also equal to the sum of the areas of the smaller squares: (a + b)^2. Expanding this expression gives:
(a + b)^2 = a^2 + 2ab + b^2
Notice that the area of the larger square can also be calculated by dividing it into smaller component shapes. It consists of the original c^2 square and four right triangles with side lengths a and b. The combined area of these triangles is 2ab.
Comparing the two expressions for the area of the larger square, we can see that they must be equal:
(a + b)^2 = c^2 = a^2 + 2ab + b^2
Subtracting 2ab from both sides of the equation gives:
c^2 = a^2 + b^2
This is the Pythagorean theorem.
Overall, the geometric proof of the Pythagorean theorem demonstrates the relationship between the areas of squares formed from the sides of a right triangle, ultimately leading to the conclusion that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.