Determine which characteristics are true of vertical angles
2 answers:
Answer:
- They are formed by intersecting lines.
- They are opposite of each other.
- They are congruent.
Step-by-step explanation:
When two lines intersect then they form two pairs of angles known as vertical angles. They are opposite to each other and have same measure.
i.e. they are congruent.
Hence, the characteristics that are true for vertical angles :
- They are formed by intersecting lines.
- They are opposite of each other.
- They are congruent.
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Answer:
5.25 m
Step-by-step explanation:
A diagram can help you understand the question, and can give you a clue as to how to find the answer. A diagram is attached. The problem can be described as finding the sum of two vectors whose magnitude and direction are known.
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<h3>understanding the direction</h3>In navigation problems, direction angles are specified a couple of different ways. A <em>bearing</em> is usually an angle in the range [0°, 360°), <em>measured clockwise from north</em>. In land surveying and some other applications, a bearing may be specified as an angle east or west of a north-south line. In this problem we are given the bearing of the second leg of the walk as ...
N 35° E . . . . . . . 35° east of north
Occasionally, a non-standard bearing will be given in terms of an angle north or south of an east-west line. The same bearing could be specified as E 55° N, for example.
<h3>the two vectors</h3>A vector is a mathematical object that has both magnitude and direction. It is sometimes expressed as an ordered pair: (magnitude; direction angle). It can also be expressed using some other notations;
- magnitude∠direction
- magnitude <em>cis</em> direction
In the latter case, "cis" is an abbreviation for the sum cos(θ)+i·sin(θ), where θ is the direction angle.
Sometimes a semicolon is used in the polar coordinate ordered pair to distinguish the coordinates from (x, y) rectangular coordinates.
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The first leg of the walk is 3 meters due north. The angle from north is 0°, and the magnitude of the distance is 3 meters. We can express this vector in any of the ways described above. One convenient way is 3∠0°.
The second leg of the walk is 2.5 meters on a bearing 35° clockwise from north. This leg can be described by the vector 2.5∠35°.
<h3>vector sum</h3>The final position is the sum of these two changes in position:
3∠0° +2.5∠35°
Some calculators can compute this sum directly. The result from one such calculator is shown in the second attachment:
= 5.24760∠15.8582°
This tells you the magnitude of the distance from the original position is about 5.25 meters. (This value is also shown in the first attachment.)
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You may have noticed that adding two vectors often results in a triangle. The magnitude of the vector sum can also be found using the Law of Cosines to solve the triangle. For the triangle shown in the first attachment, the Law of Cosines formula can be written as ...
a² = b² +o² -2bo·cos(A) . . . . where A is the internal angle at A, 145°
Using the values we know, this becomes ...
a² = 3² +2.5² -2(3)(2.5)cos(145°) ≈ 27.5373
a = √27.5373 = 5.24760 . . . . meters
The distance from the original position is about 5.25 meters.
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<em>Additional comment</em>
The vector sum can also be calculated in terms of rectangular coordinates. Position A has rectangular coordinates (0, 3). The change in coordinates from A to B can be represented as 2.5(sin(35°), cos(35°)) ≈ (1.434, 2.048). Then the coordinates of B are ...
(0, 3) +(1.434, 2.048) = (1.434, 5.048)
The distance can be found using the Pythagorean theorem:
OB = √(1.434² +5.048²) ≈ 5.248
The surface area of the actual ramp including the underside is 16.8 ft²
The height of the actual ramp = 1.2 feet
The height of the model = 6 cm
<h3>What is the surface area of a right triangular prism?</h3>A right triangular prism is a 3-dimensional triangle with two congruent triangular faces and 3 rectangular faces that connect the triangular.
The surface area of a right triangular prism
= (Perimeter of the base × Length of the prism) + (2 × Base Area)
Scale factor = 1.2 feet / 6 cm = 0.2 ft/cm
Actual base = 16 cm × 0.2 ft/cm = 3.2 ft
convert the units
9 cm = 9 cm × 0.2 ft/cm = 1.8 ft
10 cm = 10 cm × 0.2 ft/cm = 2 ft
The surface area of a right triangular prism
= (Perimeter of the base × Length of the prism) + (2 × Base Area)
Surface area of actual ramp = 2(0.5 × 3.2 ft × 1.2 ft) + 2(2 ft × 1.8 ft) + (1.8 ft × 3.2 ft)
= 16.8 ft²
Hence, the surface area of the actual ramp including the underside is 16.8 ft²
Learn more about the area here:
