determine the measures for b that will result in one nonright triangle

Question

If α = 40degress a=12 determine the measures for b that will result in one nonright triangle, no triangle, and two triangles. Round values to the nearest hundredth as needed.

A. One nonright triangle exists when b ≤

____________.

B. No triangles exist when b >

____________

.

C. Two triangles exist when ____________ < b <

____________

α = 40

Answer 1

Let c= hypotenuse, a = altitude or height (when it's a right triangle.) b = base. A = angle oppose a= 40 degrees B= the angle opposite b = 50 degrees, and C = 90 degrees = angle opposite c.

It's a right triangle when c^2 = a^2 + b^2

Use the Pythagorean Theorem: hypotenuse squared = sum of squares of the other two sides

b= a/tanA = 12/.839 = 14.30, a=12, c=18.67

18.67^2 = 12^2 + 14.30

348.49 = 144 + 204.49 = 348.49

when b= slightly over 14.30, the triangle is a right triangle

if b>14.30, it's not a right triangle, it's acute with all angles < 90

if b<14.30 it's not a right triangle, but obtuse, with one angle > 90

No triangle exists if a + b < c. There is no triangle if the sum of 2 sides < the 3rd side

b + a < c

b+12 < 18.67

if b < 6.67 then no triangle exists

if b= 6.67, the "triangle" is degenerate and collapses into two straight lines, on top of each other, colinear.

similarly if c+a < b, then no triangle exists 18.67 + 12 = 30.67 < b

IF b > 30.67, no triangle exists

no triangle exists if b< 6.67 or b > 30.67

Two triangles occur if side 6.67 < b < 30.67, neither is a right triangle

one triangle exists if b = 14.30, and then it's also a right triangle

But there's another right triangle if you treat b as the hypotenuse, with angle B as 90 degrees and C as 50 degrees. Then b = 18.67 and c = 14.30.

the hypotenuse is the side opposite the 90 degree angle. side a can't be the hypotenuse as it's opposite the 40 degree angle.

if this is a chemistry question, my guess is it has something to do with structure of molecules and the angles within them? Whatever the question is about, it's helpful to draw at least a rough sketch of the triangle(s). If it's a chemistry question, maybe get those molecule sets, chemistry kits, where you put atoms together like fitting the holes to get them connected, and notice the angles. different colors for carbon, hydrogen and other atoms.

another approach is use the law of cosines

b^2 = a^2 + c^2 - 2acCosB

14.3^2 = 12^2 + 18.67^2 -2(12)(18.67)Cos50

204.49 = 144 + 348.57 - 448.08(.643)

204.49 = 492.57 - 288.12