Chapter 8.3, Problem 19WE

a.

To determine

To complete: The statement of the theorem.

Explanation of Solution

Given:

If the square of the longest side of a triangle.......?

Calculation:

The complete statement of the theorem is:

If the square of the length of the longest side of a triangle is greater than the sum of the square of the lengths of the other two sides then the triangle is an obtuse angle.

Conclusion:

Therefore, the above theorem is converse of Pythagorean Theorem.

b.

To determine

To prove: $\Delta RST$ is an obtuse triangle.

Explanation of Solution

Given:

  $\Delta RST$ ; $l^2>j^2+k^2$ .

  

Proof:

Draw $\Delta UVW$ with legs $j$ and $k$ .

In $\Delta RST$ and $\Delta UVW$ ,

As in $\Delta UVW$

  $n^2=j^2+k^2$$\left[\text{Pythagorous Theorem}\right]$

Again in triangle both the sides j and k.

So, $l^2>j^2+k^2$

Again $n^2=j^2+k^2$

Replacing the value in the expression,

  $\Rightarrow l^2>n^2$

The length of $l$ is greater than the length of $n$ , which concludes that $\angle RST>\angle UVW$ .

Which proves that $\Delta RST$ is an obtuse triangle.

Therefore, $\Delta RST$ is an obtuse angle triangle.

Conclusion:

Therefore, $\Delta RST$ is an obtuse angle triangle with $l^2>j^2+k^2$ .