Video Transcript
For a triangle 𝐴𝐵𝐶, 𝑎 is equal
to six centimeters, 𝑏 is equal to five centimeters, and the measure of angle 𝐴 is
40 degrees. How many triangles can be
formed? Is it (A) an infinite number of
triangles, (B) no triangles can be formed, (C) one triangle, (D) two triangles, or
(E) three triangles?
In this question, we are told that
the measure of angle 𝐴 in our triangle is equal to 40 degrees. Since this is less than 90 degrees,
it is an acute angle. And as such, there are three
possibilities in terms of the number of triangles that can be formed. The number of triangles that can be
formed will be determined based on the side lengths and the height of the triangle,
which we will call ℎ.
If the side length 𝑎 is less than
the height of the triangle ℎ, then no triangles can be formed. If side length 𝑎 is equal to the
height ℎ or side length 𝑎 is greater than side length 𝑏, then one triangle can be
formed. Finally, if the height of the
triangle ℎ is less than side length 𝑎 which is less than side length 𝑏, then two
triangles can be formed.
We can therefore immediately rule
out options (A) and (E) as there is no possible way to form three triangles or an
infinite number of triangles from the measurements given.
We are told that side length 𝑎 is
equal to six centimeters and side length 𝑏 is equal to five centimeters. This means that 𝑎 is greater than
𝑏. We can therefore also rule out
option (D), as for two triangles to exist, we know that 𝑏 must be greater than
𝑎.
We are now in a position where
either zero or one triangle can be formed. In order to work out which of these
is correct, we will try to calculate the height of any possible triangle. We can do this using the right
triangle shown. The sine ratio tells us that sin 𝜃
is equal to the opposite over the hypotenuse. And in our triangle, sin of 40
degrees is equal to the height ℎ over five. Multiplying through by five, we
have ℎ is equal to five multiplied by sin of 40 degrees. This is equal to 3.213 and so
on.
Side length 𝑎 is greater than
this, as it is equal to six centimeters. We can therefore rule out option
(B), since 𝑎 is not less than ℎ. If triangle 𝐴𝐵𝐶 has side lengths
𝑎 and 𝑏 equal to six centimeters and five centimeters, respectively, and the
measure of angle 𝐴 is equal to 40 degrees, then one triangle can be formed.
Whilst it is not required in this
question, we could use the law of sines to calculate the measures of the missing
angles and side length in our triangle.
