i- ABC is an acute triangle
ii- ABC is an isosceles traingle
iii- ABC is an obtuse triangle where B is an obtuse angle
iv- ABC is a right triangle where A is the right angle
v- ABC is an obtuse triagnle where A is the obtuse angle
vi- In triangle ABC, sinA < a/c and cosA > b/c.
How many of the above statements are always false regarding triangle ABC?
iii, iv, v are NEVER true.
Here's why:
Draw the figure:
B
/|
/ |
c / |a
/ |
/ _|
A/___|_|C
b
i-ABC is an acute triangle
Yes this is always true because C is a 90° angle,
and the sum of all three angles of a triangle is
180°, so the sum of A and B must equal 90°. So each
of the other angles must be less than 90°.
ii-ABC is an isosceles traingle
This is sometimes the case when a and b are equal and
A and B are both 45°.
iii-ABC is an obtuse triangle where B is an obtuse angle
This is NEVER the case because C is a 90° angle,
and the sum of all three angles of a triangle is
180°, but an obtuse angle is greater than 90°, so
if B were an obtuse angle, that would make the
sum more than 180°.
iv-ABC is a right triangle where A is the right angle
This is NEVER the case because C is a 90° angle,
and the sum of all three angles of a triangle is
180°, and if A were 90°, that would make the sum
of A and C 180°, leaving 0° for B to equal, but
no triangle can have a 0° angle.
v-ABC is an obtuse triagnle where A is the obtuse angle
This is NEVER the case because C is a 90° angle,
and the sum of all three angles of a triangle is
180°, but an obtuse angle is greater than 90°, so
if A were an obtuse angle, that would make the
sum more than 180°.
vi- In triangle ABC, sinA < a/c and cosA > b/c.
If you use the convention where side a is opposite
angle A, side b is opposite angle B, and side c is
the hypotenuse, then this is NEVER the case because
sinA = a/c and cosA = b/c, never > or < .
However, if you vary from this notation, and have
A
/|
/ |
c / |a
/ |
/ _|
B/___|_|C
b
then sinA < a/c and cosA > a/c
Edwin
AnlytcPhil@aol.com
