Chapter 8.3, Problem 20WE

a)

To determine

To find:The complete statement of theorem 8.4.

Answer to Problem 20WE

The complete statement of theorem 8.4 is, “If the square of the longest side of a triangle is less than the sum of squares of its other two sides and if angle opposite to longest side is less than 90 degree, then the triangle is an acute triangle.”

Explanation of Solution

Giveninformation: Partial statement of theorem 8.4 is given to be completed.

Concept used:If the square of longest side in a triangle is equal to the sum of square of its other two side and angle opposite to longest side is 90 degree, then this triangle is right triangle.

Calculation:Theorem 8.4 is applied in an acute angle that says that the square of the longest side is smaller than the sum of square of its remaining two sides. Accordingly complete the statement of this theorem, starting with given partial statement.

b)

To determine

To prove: If $\Delta RST$ is an acute triangle.

Explanation of Solution

Giveninformation: Atriangle $\Delta RST$ is given in which $\overline{RT}$ is the longest side and $l^2<j^2+k^2$ .

Concept used:Use the concept of Pythagorean theorem, by drawing a right triangle $\Delta UVW$ with legs j and k. Then compare lengths l and n of both the triangle, thus formed to get the required result.

Calculation:In drawn right triangle $\Delta UVW$ , by pythagorean theorem,

  

  $n^2=k^2+j^2\mathrm{......}(i)$

But in $\Delta RST$ , it is given that,

  $\begin{array}{l}{j}^2+k^2>l^2\\ \Rightarrow n^2>l^2(by(i))\end{array}$

So, angle opposite to side l is smaller than the angle opposite to side n that is $\angle V={90}^0$ . So, $\angle S<{90}^0$ and thus $\Delta RST$ , is an acute triangle.

Conclusion: Thus it proves the statement of that, “If the square of the longest side of a triangle is less than the sum of squares of its other two sides and if angle opposite to longest side is less than 90 degree, then this triangle is an acute triangle.”