Chapter 8, Problem 14MCQ

(a)

To determine

If angle 1, 2, and 3 are congruent, find the measure of each angle.

Answer to Problem 14MCQ

  $15°$

Explanation of Solution

Given :

J designed a pinwheel to put in her garden. In the pin wheel, the blue triangles are congruent equilateral triangles, each with an altitude of 4 inches. The red triangles are congruent isosceles right triangles. The hypotenuse of a red triangle is congruent to a side of the blue triangle.

  

Calculation:

The red triangles are isosceles right angle triangles , so the measure of acute and congruent base angles is $45°$ each .

The measure of each angle of equilateral triangle is $60°$ .

Let the measure of each congruent angle 1, 2, and 3 = x.

Since the sum of measures all the central angles is $360°$ , we have :

  $\begin{array}{l}3(x)+3(45°)+3(60°)=360°\\ \Rightarrow 3x+135°+180°=360°\\ \Rightarrow 3x+315°=360°\\ \Rightarrow 3x=360°-315°\mathrm{....................}[\text{subtract }315°\text{ from each side]}\\ \Rightarrow 3x=45°\\ \Rightarrow x=15°\mathrm{....................................}[\text{divide each side by 3]}\end{array}$

So, the measure of each angle 1 , 2 , and 3 is $15°$ .

(b)

To determine

Find the perimeter of the pinwheel.

Answer to Problem 14MCQ

About 55 inches.

Explanation of Solution

Given :

J designed a pinwheel to put in her garden. In the pin wheel, the blue triangles are congruent equilateral triangles, each with an altitude of 4 inches. The red triangles are congruent isosceles right triangles. The hypotenuse of a red triangle is congruent to a side of the blue triangle.

  

Calculation:

The red triangles are isosceles right angle triangles , so the measure of acute and congruent base angles is $45°$ each .

The measure of each angle of equilateral triangle is $60°$ .

The altitude divides the equilateral triangle in $30°-60°-90°$ ratio. Let the smallest side be s .

  

  $\begin{array}{l}\cos 60°=\frac{\text{length of side adjacent to 60}°}{\text{length of hypotenuse}}=\frac{s}{AB}\\ \Rightarrow \frac{1}{2}=\frac{s}{AB}\\ \Rightarrow AB=2s\\ \sin 60°=\frac{\text{length of side opposite to 60}°}{\text{length of hypotenuse}}=\frac{4}{AB}\\ \Rightarrow \frac{\sqrt{3}}{2}=\frac{4}{2s}\\ \Rightarrow s=\frac{4}{\sqrt{3}}\end{array}$

So, the length of hypotenuse is :

  $2s=2\frac{4}{\sqrt{3}}=\frac{8\sqrt{3}}{3}$

For the red triangle , length of each leg :

  $\frac{\frac{8\sqrt{3}}{3}}{\sqrt{2}}=\frac{8\sqrt{3}}{3}\cdot \frac{1}{\sqrt{2}}=\frac{8\sqrt{3}}{3\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{6}}{6}=\frac{4\sqrt{6}}{3}$

Base of each red triangle is $\frac{8\sqrt{3}}{3}$ .

Now, find the perimeter of each wing of pinwheel:

  

The perimeter is :

  $\begin{array}{l}\frac{8\sqrt{3}}{3}+\frac{8\sqrt{3}}{3}+\left(\frac{8\sqrt{3}}{3}-\frac{4\sqrt{6}}{3}\right)+\frac{4\sqrt{6}}{3}+\frac{8\sqrt{3}}{3}\\ =\frac{8\sqrt{3}}{3}+\frac{8\sqrt{3}}{3}+\frac{8\sqrt{3}}{3}+\frac{8\sqrt{3}}{3}\\ =4\cdot \frac{8\sqrt{3}}{3}\\ =\frac{32\sqrt{3}}{3}\end{array}$

There are 3 congruent wings.

So, the perimeter of the pinwheel is :

  $3\cdot \frac{32\sqrt{3}}{3}=32\sqrt{3}\approx 55$

So, the perimeter of the pinwheel is about 55 inches.

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