3. Quilt squares are cut on the diagonal to form triangular quilt pieces. The hypotenuse of the resulting triangles is 26 inches long. What is the side length of each piece?
A. 13 inches
B. 13 √2 inches
C. 13 √3 inches
D. 26 √2 inches***
4. Find the missing value to the nearest hundredth. sin ______ = 9/16
A. 25.94 degrees
B. 29.36 degrees
C. 34.23 degrees
D. 55.77 degrees
5. Find the missing value to the nearest hundredth. cos_______ = 9/20
A. 26.74 degrees
B. 80.63 degrees
C. 63.36 degrees
D. 24.23 degrees
6. Viola drives 200 meters up a hill that makes an angle of 9 degrees with the horizontal. To the nearest tenth of a meter, what horizontal distance has she covered?
A. 1,262.8 meters
B. 200.4 meters
C. 197.5 meters
D. 31.3 meters
7. The students in Mr. Collin's class used a surveyor's measuring device to find the angle from their location to the top of a building. They also measured their distance from the bottom of the building. The diagram shows the angle measure and the distance.
(There is a large rectangle with a triangle to its right. The bottom right of the triangle says 72 degrees and the bottom of the triangle says 100 ft.)
To the nearest foot, what is the height of the building?
A. 2,400 ft
B. 72 ft
C. 308 ft
D. 33 ft
3. B. 13 √2 inches
4. B. 29.36 degrees
5. A. 26.74 degrees
6. C. 197.5 meters
7. C. 308 ft
3. To find the side length of each piece, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, we have a right-angled triangle where the hypotenuse is 26 inches long. Let's call the side lengths of each piece "a" and "b".
According to the Pythagorean theorem, we have:
a^2 + b^2 = 26^2
a^2 + b^2 = 676
Since the quilt squares are cut on the diagonal, the resulting triangles are right-angled triangles. Therefore, the two sides "a" and "b" are equal.
Plugging in "a" for both "a^2" and "b^2" in the equation above, we get:
2a^2 = 676
Dividing both sides by 2, we get:
a^2 = 338
To find "a", we take the square root of both sides:
a = √338
Approximating the square root of 338, we find that it is between 18 and 19. Therefore, the side length of each piece, "a", should be approximately 18 or 19 inches. None of the given options match this approximation, so none of the options are correct based on the information provided.
--------------------------------------------------------------------------------------------
4. To find the missing value of the angle for sin ______ = 9/16, we need to use the inverse sine function (sin^-1). The inverse sine function helps us find the angle whose sine is equal to a given value.
Using a scientific calculator or an online calculator with a sine inverse function, we can find the inverse sine of 9/16. When we do this, we get an approximate value of 34.23 degrees. Therefore, the missing value to the nearest hundredth is 34.23 degrees.
So, the correct answer is C. 34.23 degrees.
--------------------------------------------------------------------------------------------
5. To find the missing value of the angle for cos _______ = 9/20, we need to use the inverse cosine function (cos^-1). The inverse cosine function helps us find the angle whose cosine is equal to a given value.
Using a scientific calculator or an online calculator with a cosine inverse function, we can find the inverse cosine of 9/20. When we do this, we get an approximate value of 63.36 degrees. Therefore, the missing value to the nearest hundredth is 63.36 degrees.
So, the correct answer is C. 63.36 degrees.
--------------------------------------------------------------------------------------------
6. To find the horizontal distance Viola has covered, we can use trigonometry. Given that Viola drives 200 meters up a hill that makes an angle of 9 degrees with the horizontal, we can use the concept of opposite and adjacent sides in a right triangle.
In this case, the horizontal distance covered is the adjacent side, and the vertical distance covered is the opposite side. The given angle of 9 degrees is the angle between the hypotenuse (the road) and the horizontal line.
Using the trigonometric function tangent (tan), which is defined as the ratio of the opposite side to the adjacent side, we can calculate the horizontal distance.
tan(9 degrees) = (opposite side) / (adjacent side)
tan(9 degrees) = (200 meters) / (horizontal distance)
To find the horizontal distance, we rearrange the equation:
(horizontal distance) = (200 meters) / tan(9 degrees)
Using a scientific calculator or an online calculator with a tangent function, we can calculate the tangent of 9 degrees. When we divide 200 meters by the tangent of 9 degrees, we get an approximate value of 1262.8 meters for the horizontal distance.
Therefore, the correct answer is A. 1,262.8 meters.
--------------------------------------------------------------------------------------------
7. To find the height of the building, we can use trigonometry and the given angle and distance.
In the right-angled triangle formed by the surveyor's measurements, the angle of 72 degrees is opposite the height of the building, and the distance of 100 ft is the adjacent side to the angle.
Since the cosine (cos) function is defined as the ratio of the adjacent side to the hypotenuse, we can use it to calculate the height of the building.
cos(72 degrees) = (adjacent side) / (hypotenuse)
cos(72 degrees) = 100 ft / (height of the building)
To find the height of the building, we rearrange the equation:
(height of the building) = 100 ft / cos(72 degrees)
Using a scientific calculator or an online calculator with a cosine function, we can calculate the cosine of 72 degrees. When we divide 100 ft by the cosine of 72 degrees, we get an approximate value of 308 ft for the height of the building.
Therefore, the correct answer is C. 308 ft.
3. The side length of each piece can be found using the Pythagorean Theorem. The hypotenuse of the triangle is 26 inches, so the side lengths can be calculated as follows:
a^2 + b^2 = c^2
Let a and b represent the side lengths of the triangle, and c represent the hypotenuse.
a^2 + b^2 = 26^2
a^2 + b^2 = 676
Since the triangle is an isosceles right triangle (both legs are equal), we can set a = b and simplify the equation:
2a^2 = 676
a^2 = 676 / 2
a^2 = 338
a = √338
Therefore, the side length of each piece is √338 inches.
So, the correct answer is not listed.