Chapter 10.2, Problem 23WE

To determine

To construct: A square with diagonals of length $b\sqrt{2}$ .

Explanation of Solution

Given information: The figures,

  

Construction:

  Interpretation: A square with diagonal of length $b\sqrt{2}$ having sides of length b. So, a square is to be constructed with each side of length b.

At first, a straight-line l is drawn, then a point is chosen on l and is labelled as A. The compass is set for radius b. Using A as the center, an arc is drawn intersecting the line l. The point of intersection is labelled as B. Thus, a straight line is constructed of length b.

Now, all the interior angles of a square are of $90°$ .

Using A as the center and any radius, arcs are drawn intersecting l at P and Q. Using P as the center and radius greater than PA, an arc is drawn. Using Q as the center and with same radius, an arc is drawn intersecting the arc with center P at the point X. AX is drawn and is extended upward. Thus, $\angle A=90°$ is drawn.

Similarly, using B as the center and of any radius, arcs are drawn intersecting l at R and S. Using R as the center and radius greater than RB, an arc is drawn. Using S as the center and of same radius, an arc is drawn intersecting the arc with center R at the point Y. BY is drawn and is extended upward. Thus, $\angle B=90°$ is drawn.

Now, the compass is set for radius b. Using A as the center, an arc is drawn intersecting the line AX. The point of intersection is labelled as D. Using B as the center, an arc is drawn intersecting the line BY. The point of intersection is labelled as C.

The points C and D are joined. Thus, $\overline{AB}=\overline{BC}=\overline{CD}=\overline{DA}=b$ and $\angle A=\angle B=\angle C=\angle D=90°$ .

The diagonal $\overline{BD}=b\sqrt{2}$ .

Thus, a square ABCD is constructed with diagonal of length $b\sqrt{2}$ .