Video Transcript
For a triangle π΄π΅πΆ, π is equal
to two centimeters, π is equal to five centimeters, and the measure of angle π΄ is
35 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.
We will begin by trying to sketch
the triangle π΄π΅πΆ from the measurements given. We are told that the measure of
angle π΄ is 35 degrees and side lengths π and π are equal to two centimeters and
five centimeters, respectively. From the initial information, it
appears that it may be possible to sketch at least one triangle.
In order to try and prove this,
letβs recall the law of sines. This states that sin π΄ over π is
equal to sin π΅ over π, which is equal to sin πΆ over π, where uppercase π΄, π΅,
and πΆ are the measures of the three angles and lowercase π, π, and π are the
side lengths opposite them. Substituting in the measurements
given, we have sin π΅ over five is equal to sin of 35 degrees over two. We can multiply both sides of our
equation by five. This gives us that sin π΅ is equal
to 1.4339 and so on.
However, this is not possible,
since the largest value that the sine of an angle can have is one. As there are no solutions to this
equation, we can conclude that no triangles can be formed from the measurements
given. And the correct answer is option
(B).
An alternative method here would be
to calculate the height of our triangle. We know that if angle π΄ is acute
and side length π is less than the height of the triangle β, then no triangles
exist. As we have a right triangle, we can
use the sine ratio such that sin of 35 degrees is equal to the height β over
five. Multiplying through by five, we
have β is equal to five multiplied by sin of 35 degrees. And to two decimal places, this is
equal to 2.87. Since the height of the triangle,
2.87 centimeters, is greater than side length π, this confirms that no triangles
exist. And option (B) was indeed the
correct answer.
