Question

Suppose a triangle has sides a, b, and c, and that a2 + b2 < c2. Let be the measure of the angle opposite the side of length c. Which of the following must be true? Check all that apply.

A. cosθ < 0
B. the triangle is a right triangle
C. the triangle is not a right triangle
D. θ is an obtuse angle

Answer 1

A.
`Cos\theta<0`

B.The triangle is not a right triangle

C.
`\theta` is an obtuse angle

Step-by-step explanation:

We are given that a triangle with sides a, b and c.

`a^2+b^2<c^2`

When a triangle is an obtuse triangle then

`a^2+b^2<c^2`

It means given triangle is an obtuse triangle .

Obtuse triangle is that triangle in which one angle is an obtuse angle.

Let
`\theta` be the measure of angle opposite the side of length c.

We have to find the statements which are must be true.

Cosine law:

`c^2=a^2+b^2-2abCos\theta`

Substitute the value then we get

`c^2<c^2-2abCos\theta`

Subtracting
`c^2` on both sides of inequality

`c^2-c^2<c^2-c^2-2ab Cos\theta`

`0<-2ab Cos\theta`

Adding
`2abCos\theta` on both sides then we get

`2abCos\theta<-2abCos\theta+2abCos\theta=0`

`2abCos\theta<0`

`Cos\theta<0`

`Cos\theta`is negative in second quadrant and third quadrant.

Therefore,
`(\pi)/(2)<\theta<(3\pi)/(2)`

But, we know that an obtuse angle is that angle which is greater than 90 degrees and less than 180 degrees.

Hence,
`(\pi)/(2)<\theta<\pi`

Option A ,C and D are true.

Answer 2

we are given a triangle with sides corresponding to a, b and c. If this is a right triangle, then b being the longest side, the triangle's sides follow the Pythagorean theorem that states c2 = a2 + b2. In this case, the problem states a2 + b2 < c2. we can assume values and use cosine law. theta then is greater than 90 degrees which is an obtuse angle. In this case, cos theta is negative. The conditions that apply are A, C and D