Chapter 11.2, Problem 25PSB

a.

To determine

To find the measure of angleA in parallelogram AXYZ.

Answer to Problem 25PSB

The measure of angle Ais ${30}^{\circ }$ .

Explanation of Solution

Given:

  $\begin{array}{l}AZ=8,\\ XN=6,\\ XM=4.\end{array}$

Formula Used:

The distance between two points by using the distance formula, which is an application of the Pythagorean theorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

The below theorem is used:

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  $\begin{array}{l}Areaoftriangle=\frac{1}{2}\ast b\ast h\\ b=Base,\\ h=Height\end{array}$

Calculation:

  

In a parallelogram, opposite sides are congruent.

  $\begin{array}{l}\overline{AZ}=\overline{XY}\\ XY=8\end{array}$

In right angled $\Delta XMY$ , we get

  ${\left(XY\right)}^2={\left(XM\right)}^2+{\left(MY\right)}^2$

  $\begin{array}{l}{\left(8\right)}^2={\left(4\right)}^2+{\left(MY\right)}^2\\ {\left(MY\right)}^2=64-16\\ MY=\sqrt{48}=4\sqrt{3}\end{array}$

Since the sides of the triangle are $4,4\sqrt{3},8$ , it must be a ${30}^{\circ }{60}^{\circ }{90}^{\circ }$ triangle with $\angle Y={30}^o$ .

In a parallelogram, opposite sides are congruent.

  $m\angle A={30}^o$

b.

To determine

To compute the length of side AX in parallelogram AXYZ.

Answer to Problem 25PSB

The length of side AX is $12$ .

Explanation of Solution

Given:

  $\begin{array}{l}AZ=8,\\ XN=6,\\ XM=4.\end{array}$

Formula Used:

  $\begin{array}{l}Areaofparallelogram=b\ast h\\ b=Base,\\ h=Height\end{array}$

Calculation:

  

  $\begin{array}{l}A(▱AXYZ)=AZ\times XN\\ =8\times 6\\ A(▱AXYZ)=48\\ A(▱AXYZ)=AX\times MX\\ 48=AX\times 4\\ AX=\frac{48}{4}=12\end{array}$