Can you answer this question?

Question

These are some of the items on my desk: a red plastic triangle, a sheet of white paper, and a sharpened pencil. The plastic triangle (ABC) has an angular arrangement of 30-60-90 (CAB-ACB-ABC). The segment (CB -or- BC) whose endpoints are the vertex 60 and vertex 90 is 3.5 feet long. What are the lengths of the other two sides (provide the proof)?

My bad. The intermediate algebra course was my intended victim. I want an analytical geometry response. I don't understand trigonmetry well-enough (dropped for sex on wednesdays; all about a unit circle thing.)

If analytical geometry shows up, then leave a response.

Using Trigonometry to solve this question sure would make math simpler, but this is how I solved the problem. What I wrote is questionable, but the solution is correct.

The named points (and angle definitions) of the triangle were ABC (60-90-30; where each named point corresponds to an angle definition). Using symmetry, I can add a point D to the graph as a segment BD. This new point can be used to reflect the original triangle such that a distance to the coordinate D is equidistant to B as C (is to B). Similarly, coordinate A should be equidistant to point D as A is to C. The new (and larger) triangle is formed. Segment AB is an “angle bisector” for Triangle ADC. Anyway, the new triangle is an Equilateral Triangle (60-60-60) because the sum of the altitude angles (30 + 30) equal 60; also, the 90 degree is often unimportant at this point. Therefore, 60 degrees at the altitude and 60 degrees (each) at the isocelic base.

Here is the point: we can use Pythagorean Theorem. Earlier, I wrote about a new triangle where the angles equal 60. Also, I know that each side of the triangle is of equal length. If CB (original question) is 3.5 feet, and BD is 3.5 feet (or 7/2 feet), then the sum of the 2 segments is 7 feet (or 14/2 feet). So each side of the large equilateral triangle is 7 feet. The altitude [segment AB] for Triangle ABC is still unknown!

Pythagorean Theorem states that the square of the hypoteneuse of a Right Triangle is a result of the sum of the square of the altitude and the square of the base, such as:
(b^2) + (a^2) = (h^2)

The proof of a function (as Pythagorean’s Theorem):

((7/2)^2) + (a^2) = (7^2)

(49/4) + (a^2) = 49

-(49/4) + (49/4) + (a^2) = 49 - (49/4)

(a^2) = (4/4)*49 - (49/4)

(a^2) = (196/4) - (49/4)

(a^2) = (147/4)

a = SQRT(147/4)

a = SQRT(3 * 49/2 * 2)

a = +/- 7 SQRT(3)/2

a = 7 SQRT(3)/2; or roughly 6.06

*SQRT means square root (or radical of a number).

You know what? Let's just vote.

Answer 1

AC = 3.5/Sin(30) and AB = 3.5/Tan(30). What's to prove?

Answer 2

You can use the law of sines for this quite easily.

a/sinA = b/sinB = c/sinC

Since CB is across from 30, we say 3.5/sin(30) = y/sin(60) = z/sin(90)

sin(30) = .5 and sin(90) = 1, so that's easy.
3.5/.5 = 7/1

7/1 = y/sin(60)
sin(60) * 7 = y
y = 6.06 (actually it's 7 * sqrt(3)/2)

This is a known type of triangle where the sides are 1, sqrt(3)/2, and 2