In the triangle ABC what is the length of BC if the angle A equals sixty degrees AB equals six cm and AC equals four cm?

Answer 1

Approximately 5.29 centimeters. There may be an easier way to figure it out, but this is what I came up with. Draw your triangle on a piece of paper. Because you don't know at this point whether it's an acute triangle (all angles < 90 degrees), a right triangle (one angle = 90 degrees), or an obtuse triangle (one angle > 90 degrees), you will want to draw all three possibilities. But when you try to draw these triangles, you will be able to tell that IF any angle is > or = 90 degrees, it will HAVE to be angle C. Starting with the acute triangle: draw a line from point B to a point, somewhere in the middle of line AC, that forms a 90 degree angle with line AC. Label this point D. Triangle ABD is now a right triangle, with a right angle at angle D. Now, there are all kinds of things you can figure out about this triangle, some of which may help you figure out things about the original triangle ABC. One of the things we know about any right triangle is that, if, in addition to the 90-degree right angle, we know the measurement of at least one other angle, and at least one side, we can figure out all the sides and all the angles. To do so, we need to use the trigonometric functions - sine, cosine, tangent, etc. Now, we can look at a table of trigonometric functions and easily determine the sine, cosine, etc. of any angle, provided we know the measurement of that angle. The angle we know is angle A, 60 degrees. But which function do we want to use here? The definition of cosine is adjacent over hypotenuse, or in other words, the length of the side adjacent to the angle (but not opposite the right angle) divided by the length of the side that is opposite the right angle. In this case, line AB is the hypotenuse, because the right angle is at D. Line AD is the adjacent side. We know the length of the hypotenuse, but not of the adjacent side. But that's okay because we can easily figure it out. The formula is cos A = AD/AB. Now substitute what we know - cos 60 = AD/6. Then look up the cosine of 60 degrees in a trig table, and you get 0.5, so 0.5 = AD/6. Solving for AD yields AD = 3. At this point, we need to look back at our 3 different versions of the triangle ABC. We know now that AD(3 cm) is shorter than AC (4 cm), and therefore we were right to put point D somewhere between A and C. If AD had been longer than AC, then, for visual purposes, we would want to deal with the obtuse triangle rather than the acute one. What you would do in this case is extend line AC past C until you can draw a line from point B down to the extended line AC to form a right angle, then label this point D. But either way, we can now figure out the length of line CD. If AC is longer than AD (as is the case here), then CD = AC - AD = 4 - 3 = 1. If AD is longer than AC, then CD = AD - AC. Oh, if AD is exactly equal to AC, then triangle ABC was a right triangle all along, and we can easily figure out AC by using the sine function. Now, going back to the acute triangle we drew earlier, since we now know that ABC is, in fact, an acute triangle. Not only do you have a right triangle in ABD, you have another right triangle defined by BCD. The Pythagorean theorum gives us an easy method for determining the length of any side given the other two sides. But we only know one side of BCD, CD = 1. Can we figure out BD so that we have the necessary two sides' lengths? Yes, we can, if we go back to right triangle ABD, which shares a side (BD) with BCD. The sine of an angle is defined as the opposite divided by the hypotenuse. In this case, the opposite is line BD, which we do not know, and the hypotenuse is, again, AB, which we know to be 6. And, from the trig tables, we know that the sine of angle A (60 degrees) is, approximately, 0.866. The formula is sin A = BD/AB, or, substituting what we know, 0.866 = BD/6, then solving for BD yields BD = (app) 5.196. So, now we know both BD and CD, the two non-hypotenuse sides of right triangle BCD. The remaining side, the hypotenuse, is BC, which is the same line we set out to determine the length of. Now we use the Pythagorean theorum, which says that the squares of the lengths of the two non-hypotenuse sides of a right triangle, added together, exactly equal the square of the length of the hypotenuse. Or a2 + b2 = c2, where c is the length of the hypotenuse, and a and b are the length of the other two sides. (Don't get confused with the capital letters A, B, and C, which you used to designate the vertices of the original triangle. The lower case letters here represent the lengths of the sides of triangle BCD.) So, now, if a = CD = 1, and b = BD = 5.196, then a2 + b2 = 12 + 5.1962 = 1 + 27 = 28. But this is equal to c2. So, to solve for c, just take the square root of 28, which is approximately equal to 5.29. Though the question is answered now, I would like to expand the answer a little bit. Every triangle has three sides and three angles, for a total of 6 "pieces". It turns out that, if you know the measurements of any three of these pieces, and at least one of them is a side, you can determine the measurements of the other three pieces. We started with three pieces (two sides and an angle), and since at least one of them is a side, we know we can get the rest. We already determined the length of the third side. All that is left to completely define this triangle is the measurements of angles B and C. These can be obtained as well. We'll start with angle C. Remember that BCD is a right triangle, so we can use the trig functions again. The cosine of angle C, with respect to the right triangle BCD, is defined as adjacent divided by hypotenuse, or CD/BC. In this case, we know both CD and BC, but we do not know the measurement of angle C. But we can compute the value of cos C, which is CD/BC = 1/5.29 =(app) 0.189. Now that we know this, we can use the trig tables in a backwards manner. Instead of looking up an angle, we'll find the value 0.189 in the body of the table and trace it to the axes of the table to find the measurement of the angle. This is called finding the "inverse cosine" of a value, or in other words, the angle measurement whose cosine is equal to the given value. This can also be done on a scientific calculator. The inverse cosine of 0.189 is, approximately, 79 degrees. We now have the measurement of angle C. For angle B, the job is much easier. We know that the sum of all three angle measurements in any triangle is 180. Thus, computing the measurement of angle B is simply a matter of subtracting the measurements of angles A and C from 180. B = 180 - A - C = 180 - 60 - 79 = 41. And now we have the measurement of angle B. In summary, the 6 pieces of triangle ABC are (with the ones we computed in bold): * Angle A 60o

* Side AB 6 cm

* Angle B 41o

* Side BC 5.29 cm

* Angle C 79o

* Side AC 4 cm