- #1
skrat
- 748
- 8
Homework Statement
Side ##a=4 cm##, altitude to side a is 3 cm , angle ##\alpha =60 °##.
How can I draw that? Step by step
I didn't read any further.Mentallic said:If we draw side a=4cm alone the bottom, let A be the left point, B the right point (hence AB is 4cm)...
Not really. :DMentallic said:You'll be surprised with how much time is spent wasted because the problem was misunderstood and the standards of triangle labelling aren't always upheld.
Mentallic said:You can use the sine and cosine rule to find length c. If b = 3cm then with the cosine rule
[tex]c^2 = a^2+b^2-2ab\cdot \cos C[/tex]
Plugging our known values in
[tex]c^2=3^2+4^2-2\cdot 3\cdot 4\cos C[/tex]
[tex]c^2=25-24\cos C[/tex]
Now for the sine rule
[tex]\frac{\sin A}{a} = \frac{\sin B}{b}[/tex]
Plugging values
[tex]\sin B = \frac{3\sqrt{3}}{8}[/tex]
Hence you've got B and then you can find [itex]\sin C[/itex]. I'm assuming you're only using a compass, because if you could use a protractor then just the sine rule would be sufficient to solve your problem.
Yes but it is a waste of time in a good way. :DMentallic said:Ugh... altitude = 3cm... I took that as being the side length of the triangle. Sorry, it's been a long night. That's another thing that'll waste everyone's time
YES.Mentallic said:Just once more... Altitude to side a means that the altitude of 3cm connects point A to side a perpendicularly, correct?
Mentallic said:If that is the case (which at this point I wouldn't bet heavily on), you have many possibilities to choose from. It could be an isosceles triangle with each half-triangle having side lengths [itex]2,3,\sqrt{13}[/itex] by using pythagoras' theorem, which would then be easy to construct.
skrat said:I was afraid you might say that, because I came to the same conclusion (but only in my head with no proof). It's not that I have to proof anything but it just came to my mind that it might be interesting to see the mathematical proof of more possibilities and with it also exactly what are they.
Mentallic said:Actually we should have thought about it some more, because it doesn't look as though it's going to have many possible answers. If you want to prove it for yourself then denote the angle at A in the left triangle by [itex]\theta[/itex] and then the angle at A in the right triangle will be [itex]60^o-\theta[/itex]. Using simple trigonometry you can then find the length of side a on only the left triangle (you can denote it by a1) and similarly, on only the right triangle (denoted [itex]a_2=4-a_1[/itex]).
You should then have two equations in two unknowns.
AlephZero said:You can construct it geometrically if you know this theorem: http://www.mathopenref.com/arccentralangletheorem.html
Draw side a, find the center of the circle and draw it, then draw a line parallel to a at the given altitude.
Mentallic said:I didn't see that at first. You must have edited your post, right? Ugh this is too embarrassing, good night guys.
MrAnchovy said:Brilliant - did you get this by guessing that the angle of 60° was chosen because 60/2 = 30 and 60 + 30 = 90?
MrAnchovy said:Brilliant - did you get this by guessing that the angle of 60° was chosen because 60/2 = 30 and 60 + 30 = 90?
To draw a triangle with these measurements, you will need a ruler, protractor, and a pencil. First, draw a line segment of 4 cm on a piece of paper. Then, using the protractor, draw an angle of 60° from one end of the line segment. This will be the base of the triangle. Next, from the other end of the line segment, draw a perpendicular line of 3 cm. This will be the altitude of the triangle. Finally, connect the end points of the altitude to the end point of the angle to form the triangle.
The formula for finding the area of a triangle is: Area = 1/2 * base * height. In this case, the base is 4 cm and the height is 3 cm, so the area would be: Area = 1/2 * 4 cm * 3 cm = 6 cm²
To find the missing side of a triangle, you can use the Pythagorean theorem, which states that the sum of the squares of the two shorter sides is equal to the square of the longest side. In this case, we can use the altitude as the longest side, so we can set up the equation as: 3² = 4² + x². Solving for x, we get: x = √(3² - 4²) = √(9 - 16) = √(-7) = undefined. Since the square root of a negative number is undefined, this triangle cannot be drawn as it violates the triangle inequality theorem.
Yes, you can draw a triangle with these measurements on a coordinate plane. To do so, start by plotting a point at (0,0) and label it as A, this will be the vertex of the 60° angle. Then, plot another point at (4,0) and label it as B, this will be one end of the base. From point A, draw a line at a 60° angle to the x-axis, and mark a point at (3,3√3) and label it as C, this will be the other end of the base. Finally, connect points B and C to complete the triangle.
The perimeter of a triangle is the sum of all its sides. In this case, we can use the Pythagorean theorem to find the length of the missing side, which we found to be undefined in question 3. Therefore, the perimeter would be: Perimeter = 4 cm + 3 cm + undefined = undefined. Again, this triangle cannot be drawn as it violates the triangle inequality theorem.