A student chooses two distinct real numbers x and y with 0 < x < y < 1.
The student then attempts to draw a triangle ABC with AB = 1, sin A = x, sin B = y
Which of the following statements is/are correct?
I For some choice of x and y, there is exactly one triangle the student could draw.
II For some choice of x and y, there are exactly two different triangles the student could draw.
III For some choice of x and y, there are exactly three different triangles the student could draw.
(Note that congruent triangles are considered to be the same.)
- Anone of them
- BI only
- CII only
- DIII only
- EI and II only
- FI and III only
- GII and III only
- HI, II and III
Show the answer and worked solution
answer · C
- Anone of them
- BI only
- CII only
- DIII only
- EI and II only
- FI and III only
- GII and III only
- HI, II and III
Write α = arcsin x and β = arcsin y, both acute, with α < β because x < y. Each of A and B can be acute or obtuse, giving four candidate pairs, and the only test is whether A + B < 180∘. Both acute always works. Making A obtuse gives A + B = 180∘ − α + β, which fails since β > α. Making B obtuse gives 180∘ − β + α < 180∘, which works. Both obtuse is impossible. So there are always exactly two triangles, whatever x and y are: II is correct and I and III never happen.