Chapter 14.4, Problem 7E

Solution

To find:The correct statement/option for constructing a triangle.

The case (C) is ambiguous.

Given data: A triangle can be constructed with compass and straightedge when, given three parts of the triangle which one of the following would not result a required triangle, i.e., which is the ambiguous case:

  $\begin{array}{l}\left(A\right)\text{Given:}\text{three}\text{sides}\\ \left(B\right)\text{Given:}\text{two sides}\text{and}\text{one}\text{included}\text{angle}\\ \left(C\right)\text{Given:}\text{two}\text{sides}\text{and}\text{one}\text{non-included}\text{angle}\\ \left(D\right)\text{Given:}\text{two}\text{angles}\text{and}\text{one}\text{non-included side}\end{array}$

Method/Formula used:

The triangle is tried to obtain from given methods ( A ), ( B ), ( C ), and ( D ).

Calculations:

(A): When three sides AB, BC, and CA are given the triangle is drawn as follows:

Sketch side AB, set length BC in the compass, then draw arc of radius of length BC taking B as its center. Then set length AC in the compass and draw an arc of radius AC taking A as the center. The point of intersection of two arc is vertex C of the triangle and ABC is the required triangle.

  

(B): When two sides AB, AC, and included angle CAB is given then triangle is drawn as follows:

Draw lines AB and AC making angle CAB, then join AC so ABC is the required triangle.

  

(C): When two sides AB, AC, and non-included angle ABC is given then triangle is drawn as follows:

Draw lines AB and AC making an arbitrary (of your choice).Then to make angle ABC of required value, the line from B may or may not join at C, it may intersect line AC at any point C or C’ or C” so the triangle may not be the required triangle.

  

Therefore, the case (C) is ambiguous.

(D): When two angles ABC, ACB, and included side BC is given then triangle is drawn as follows:

Draw angles MBC and NCB on given side BC, then point of intersection of BM and CN is A so ABC is the required triangle.

  

Thus, the case (C) is ambiguous.