Chapter 12, Problem 34CR

a.

To determine

To find: Theside AE in a parallelogram ABCD.

Answer to Problem 34CR

The side AEis$8$ .

Explanation of Solution

Given information:

Side $AB=10,$

Side $AF=12,$

Side $AD=15.$

Formula used:

The below property is used:

The corresponding sides of similar triangles are congruent.

Calculation:

  

Opposite angles in a parallelogram are congruent.

  $\angle B\cong \angle D$

ΔAEB is similar to ?AFD by AA similarity rule.

The corresponding sides of similar triangles are congruent.

  $\begin{array}{l}\frac{AB}{AD}=\frac{AE}{AF}\\ \frac{10}{15}=\frac{AE}{12}\\ AE=\frac{120}{15}\\ AE=8\end{array}$

b.

To determine

To calculate: The ratio of x and y.

Answer to Problem 34CR

The ratio of x and yis$9:1$ .

Explanation of Solution

Given information:

Side $AB=10,$

Side $AF=12,$

Side $AD=15.$

Formula used:

The below property is used:

The corresponding sides of similar triangles are congruent.

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”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Calculation:

  

Opposite angles in a parallelogram are congruent.

  $\angle B\cong \angle D$

ΔAEB is similar to ?AFD by AA similarity rule.

The corresponding sides of similar triangles are congruent.

  $\begin{array}{l}\frac{AB}{AD}=\frac{AE}{AF}\\ \frac{10}{15}=\frac{AE}{12}\\ AE=\frac{120}{15}\\ AE=8\end{array}$

Side BE can be calculated by applying Pythagoras Theorem.

In right angled triangle AEB, we get

  $\begin{array}{l}{\left(AB\right)}^2={\left(AE\right)}^2+{\left(EB\right)}^2\\ {\left(10\right)}^2={\left(8\right)}^2+{\left(EB\right)}^2\\ {\left(EB\right)}^2=100-64\\ {\left(EB\right)}^2=36\\ EB=\sqrt{36}=6\end{array}$

  $\begin{array}{l}\overline{AD}\cong \overline{BC}\\ AD=15,\\ BC=15.\\ x=BC-BE\\ x=15-6=9\end{array}$

Side FD can be calculated by applying Pythagoras Theorem.

In right angled triangle ADF, we get

  $\begin{array}{l}{\left(AD\right)}^2={\left(AF\right)}^2+{\left(DF\right)}^2\\ {\left(15\right)}^2={\left(12\right)}^2+{\left(DF\right)}^2\\ {\left(DF\right)}^2=225-144\\ {\left(DF\right)}^2=81\\ DF=\sqrt{81}=9\end{array}$

  $\begin{array}{l}\overline{AB}\cong \overline{DC}\\ AB=10,\\ DC=10.\\ y=DC-FC\\ y=10-9=1\end{array}$

The ratio of x and y is $9:1$ .