Chapter 6.3, Problem 14WE

To determine

To prove: $\angle ACB$ is not a right angle.

Explanation of Solution

Given information:

  $AT=BT=5$ ; $CT=4$

Given figure,

  

Calculation:

Assume temporarily that $\angle ACB$ is a right angle.

By the property of right angles triangles,

  $\begin{array}{c}A{B}^2=A{C}^2+B{C}^2\\ {10}^2=6^2+8^2\end{array}$

There can be only 2 cases:

  $AC=6,BC=8\text{or}AC=8,BC=6$

Two triangles with two of its sides of dimension 4 and 5 cannot have the third side of a different dimension.

Hence, initial assumption that $\angle ACB$ is a right angle is incorrect.

Therefore, it is proved that $\angle ACB$ is not a right angle.