If a right triangle has an angle that is 65°

Question

65
°
, what is the third angle in the triangle?(1 point)
°

2. What is the measure of the exterior angle x
𝑥
? (1 point)

Angle inside the triangle: 37°

Angle inside the triangle: 42°

Unknown angle: x°

The diagram shows a triangle with two known angles (37° and 42°) and an unknown angle (x°) adjacent to the 42° angle.

3. Use the figure to answer the question.A rightward inclined transversal passes through two horizontal parallel lines labeled upper S and upper T. Angles formed at the intersection point of line upper S are labeled clockwise from the top: 1, 2, 4, and 3. Corresponding angles formed at the intersection point of line upper N are labeled clockwise from the top: 5, 22 degrees, 7, and 6.In the picture, one angle is labeled 22° and lines S and T are parallel. Which other three angles will be 22° ? Separate each angle number with a comma.(1 point)angles Skip to navigation

4. Use the image to answer the question.A triangle has the vertices labeled upper M upper N upper O.
What is the relationship of the sides in this triangle based on the Pythagorean Theorem?

(1 point)
m^2+n^2=

5. If a triangle has sides measuring 60, 75, and 45, is it a right triangle?(1 point)
Responses

Yes, because 5,625 equals 5,625.
Yes, because 5,625 equals 5,625.

No, because 9,225 does not equal 2,025.
No, because 9,225 does not equal 2,025.

No, because 5,625 does not equal 3,600.
No, because 5,625 does not equal 3,600.

Yes, because 3,600 equals 3,600.

6. If triangle DEF
𝐷
𝐸
𝐹
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.(1 point)

7. If a triangle has sides of length 12 and 16, how long is the hypotenuse?(1 point)

8. If a right triangle has a leg with length 12 and the hypotenuse is 16, how long is the other leg? Round to the nearest tenth, if necessary.(1 point)

9. Use the image to answer the question.An illustration shows a square with smaller squares representing baseball plates on the top left, top right, and bottom right vertices. The bottom left vertex has a pentagon that represents home plate. A stick figure is located on the top of the square and the measurement from the right vertex to the stick figure is labeled 30 feet. A dashed diagonal line runs from the stick figure to the inner right side of the square, forming the hypotenuse of a triangle. The top right side of the square forms the right angle of the triangle and is marked with a right angle symbol. The outer right side of the square is labeled 90 feet. 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.(1 point) feet

10. It depicts a Cartesian coordinate system.

There is a line segment connecting two points:
(
3
,
2
)
and
(
7
,
8
)
.

The x-axis ranges from 0 to 9.

The y-axis also ranges from 0 to 9.

The line segment is drawn in orange.

The coordinates of the points are marked with black dots and labeled next to the dots.

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth.

Answer 1

Sure! Here are the answers to each of your questions:

1. Third Angle in a Right Triangle: In a right triangle, the sum of the angles is always 180°. Since one angle is 90°, and another is 65°, the third angle can be calculated as follows: \[ 180° - 90° - 65° = 25° \] Answer: 25°

2. Measure of Exterior Angle \( x \): The sum of the angles in a triangle is also 180°. Given that the two known angles are 37° and 42°, we can find the third angle \( x \): \[ 37° + 42° + x = 180° \] \[ x = 180° - 37° - 42° = 101° \] Answer: 101°

3. Corresponding Angles: Since lines S and T are parallel and an angle of 22° is formed at the intersection with the transversal, the corresponding angles formed at the other intersection (angles 5, 6, 7) will also be equal to 22°. Answer: 5, 7, 6

4. Relationship of the Sides in Triangle \( \Delta MNO \) based on the Pythagorean Theorem: The Pythagorean Theorem states that for a right triangle: \[ a^2 + b^2 = c^2 \] If we consider the sides \( m \) and \( n \) to be the legs, and the hypotenuse as \( c \): Answer: \( m^2 + n^2 = c^2 \)

5. Is the triangle with sides 60, 75, and 45 a right triangle?: We check using the Pythagorean Theorem:

   • \( 60^2 + 45^2 = 3600 + 2025 = 5625 \)
   • \( 75^2 = 5625 \) Since \( 60^2 + 45^2 = 75^2 \), it is a right triangle. Answer: Yes, because 5,625 equals 5,625.

6. Is triangle DEF with sides 50, 40, and 32 a right triangle?: We check:

   • \( 50^2 = 2500 \)
   • \( 40^2 + 32^2 = 1600 + 1024 = 2624 \) (not equal)
   • Check the largest triangle side to see: \( 50^2 = 40^2 + 32^2 \) Therefore, the triangle is NOT a right triangle. Answer: 0

7. Length of the Hypotenuse: For a triangle with legs 12 and 16, we use: \[ c = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \] Answer: 20

8. Length of the Other Leg when one leg is 12 and the hypotenuse is 16: \[ a^2 + 12^2 = 16^2 \] \[ a^2 + 144 = 256 \] \[ a^2 = 256 - 144 = 112 \] \[ a = \sqrt{112} \approx 10.583 \] Rounded to the nearest tenth: Answer: 10.6

9. Distance for the throw from the shortstop to first base: The horizontal distance (base path) is 90 feet, and she is 30 feet from second base. We can visualize this as a right triangle: \[ d = \sqrt{(90)^2 + (30)^2} = \sqrt{8100 + 900} = \sqrt{9000} \approx 94.87 \] Rounding to the nearest tenth: Answer: 94.9 feet

10. Length between the two points using the Pythagorean Theorem: For points (3,2) and (7,8), calculate: \[ \text{Length} = \sqrt{(7-3)^2 + (8-2)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21 \] Rounded to the nearest hundredth: Answer: 7.21

If you have any more questions or need further explanations, feel free to ask!

Answer 2

11. Use the image to answer the question.An illustration shows a square pyramid with labeled parts. The front and left faces of the pyramid and base are visible. The back and right faces are not visible. The back and right base edges are not visible but their position is indicated by a dashed line. The left base edge shows a length of 16 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and shows a height of 15. A right angle is indicated where the dotted height line meets the base.Find the slant height of the square pyramid.(1 point) units

12. Use the image to answer the question.An illustration of a rectangular prism is shown with labeled parts. The top, front and right sides are visible. The bottom, left, and rear sides are not visible, but their positions are indicated by dashed lines. The vertical edge joining the front and left sides shows a height of 12 centimeters. The front base edge shows a width of 3 centimeters. The right base edge shows a length of 4 centimeters. A dotted line extends diagonally from the vertex at the bottom back right side of the prism through the interior of the prism to the vertex at the top front left.Find the length of the diagonal of the prism.(1 point)cm

13. The Great Pyramid of Giza in Egypt has a square base with edge length of 230.5 m and a slant height of 180.2 m. Find the height of the pyramid. Round the answer to the nearest tenth.(1 point)
m

14. 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 each diagonal. Round the answer to the nearest tenth.(1 point)
in.

Answer 3

Let’s find the answers to each of your new questions:

11. Slant Height of the Square Pyramid:

For a square pyramid, the slant height can be found using the height of the pyramid and half the length of the base. The formula can be expressed as:

\[ \text{slant height} = \sqrt{h^2 + \left(\frac{b}{2}\right)^2} \]

where \(h\) is the height (15 feet) and \(b\) is the base edge length (16 feet).

Calculating:

\[ \text{slant height} = \sqrt{15^2 + \left(\frac{16}{2}\right)^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \text{ ft} \]

Answer: 17 ft

12. Length of the Diagonal of the Prism:

To find the length of the diagonal in a rectangular prism, we can use the formula:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where \(l\) is the length (4 cm), \(w\) is the width (3 cm), and \(h\) is the height (12 cm).

Calculating:

\[ d = \sqrt{4^2 + 3^2 + 12^2} = \sqrt{16 + 9 + 144} = \sqrt{169} = 13 \text{ cm} \]

Answer: 13 cm

13. Height of the Great Pyramid of Giza:

To find the height of the pyramid, we can use the relationship between the slant height, height, and half the base length. The formula is:

\[ s^2 = h^2 + \left(\frac{b}{2}\right)^2 \]

where \(s\) is the slant height (180.2 m), \(b\) is the base edge length (230.5 m), and \(h\) is the height we want to find.

Calculating:

\[ h^2 = s^2 - \left(\frac{b}{2}\right)^2 \]

First, calculate \( \frac{b}{2} \):

\[ \frac{b}{2} = \frac{230.5}{2} = 115.25 \text{ m} \]

Now substituting:

\[ h^2 = 180.2^2 - 115.25^2 = 32472.04 - 13273.0625 = 19198.9775 \] \[ h = \sqrt{19198.9775} \approx 138.645 \] Rounding to the nearest tenth:

Answer: 138.6 m

14. Length of Each Diagonal of the Coffee Table:

To find the diagonal of a rectangular prism (which in this case is the coffee table), we use the same formula as above:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

Here, the dimensions are: length \(l = 44\) in, width \(w = 24\) in, and height \(h = 14\) in.

Calculating:

\[ d = \sqrt{44^2 + 24^2 + 14^2} = \sqrt{1936 + 576 + 196} = \sqrt{2708} \approx 52.0 \text{ in} \]

Rounding to the nearest tenth:

Answer: 52.0 in

If you have more questions or need further assistance, please let me know!

Answer 4

2. 79
3. 2,3,6
4. o^2 not zero 0 its the letter o
13. 138.5

Answer 5

Thank you for your responses! Let’s review your answers to ensure they are accurate and consistent with the calculations:

Question Review

2. Measure of angle \( x \): You had an initial answer of 101° calculated based on the given angles in a triangle (37° + 42°). If your new answer is 79°, it doesn't seem mathematically justified in this context. Please check the setup of the angle relationships.

3. Corresponding Angles: You provided the angles 2, 3, 6. If angle 2 is directly corresponding to angle 22° when the lines are parallel, then that is accurate. The previously calculated answer of 5, 7, 6 would depend on their spatial relationship. Please double-check the diagram for confirmation.

4. Pythagorean Theorem Setup: If your answer for the relationship of the sides in the triangle is \( o^2 \), that means you're referring to the sides of a triangle where \( o \) represents the length of one of the sides. This means the relationship should be set up as \( m^2 + n^2 = o^2 \) where \( o \) is the hypotenuse.

5. Height of the Great Pyramid of Giza: You indicated 138.5 m as the height. According to the prior calculation, the height derived was approximately 138.6 m, but rounding can vary slightly. However, if you are rounding differently, both could be considered correct depending on the rounding convention.

If any part of my interpretations seems off based on your additional information, or if you would like further clarification or recalculations, just let me know!