Begin with recalling that the sum of the interior angle measures of an n-sided convex polygon is (n-2)180.
Yes, see solution.
Practice makes perfect
We are asked to determine if the floor of the gazebo, that is a regular hexagon, will be enough room for Eva and her guests. Let's redraw the given picture and name the vertices with consecutive letters.
Our first step will be to evaluate the interior angles of this hexagon. To do this, let's recall the Polygon Interior Angles Sum Theorem.
The sum of the interior angle measures
of an n-sided convex polygon is(n-2)180.
We will find the sum of the interior angle measures of a hexagon by substituting 6 for n in the above formula.
Since the given polygon is regular, all interior angles are congruent. This means that to find the measure of each of the interior angles, we need to divide the sum of all interior angles, 720, by the number of sides, which is 6.
720/6=120
Each interior angle of this polygon measures 120. From now on, we will focus on â–³ AFG as all four drawn triangles are congruent.
Notice that ABDE is a rectangle. This means that ∠EAB is a right angle and the measure of ∠GAF is the difference between 120 and 90.
m∠GAF=120-90=30
From the properties of a regular hexagon, we know that segment FC is a bisector of ∠F as it divides the hexagon into two congruent isosceles trapezoids. Therefore, the measure of ∠GFA is half the measure of an interior angle.
m∠GFA=1/2(120)=60
Let's add this information to our picture. As we can see, △ AGF is a 30^(∘)-60^(∘)-90^(∘) triangle. Remember, as this hexagon is regular, it has all sides of the length of 7 ft.
Recall that, according to the 30^(∘)-60^(∘)-90^(∘) Triangle Theorem, the length of the hypotenuse, FA, is two times the length of the shorter leg, FG.
FA= 2 * FG
7=2* FG ⇒ FG=3.5
The length of FG is 3.5. Next, from this theorem we know that the longer leg, AG, is sqrt(3) times the length of the shorter leg, FG.
AG=sqrt(3)FG
AG=sqrt(3)( 3.5)=3.5sqrt(3)
The length of AG is 3.5sqrt(3). Now we can add this information to our picture.
Next, we can evaluate the area of â–³ AGF using the fact that the area of a right triangle is the product of its legs multiplied by 12.
A_(AFG)=1/2* 3.5* 3.5sqrt(3)
⇓
A_(AFG)=6.125sqrt(3)≈ 10.6
The area of â–³ AGF is approximately 10.6 square feet. Since all four triangles are congruent, then they have the same areas.
As we can see, to find the area of the floor of the gazebo we need to find the area of a rectangle. Since all the triangles are congruent, we can evaluate the length of AE by multiplying the length of AG by 2.
AE=2AG
AE=2(3.5sqrt(3))=7sqrt(3)
With this, we can evaluate the area of rectangle ABDE by multiplying its side lengths.
A_(ABDE)=AB* AE
A_(ABDE)=7*7sqrt(3)=49sqrt(3)≈ 84.8
The area of ABDE is approximately 84.8 square feet.
Finally, we will evaluate the area of the floor of the gazebo by adding the area of a triangle multiplied by 4 and the area of a rectangle.
A_(ABCDEF)=4* 10.6+84.8=127.2
The area of the floor of the gazebo is approximately 127.2 square feet. We know that Eva wants to allow 8 square feet of area for each guest. Therefore, by dividing the area of the floor of the gazebo by 8 we will find the maximum number of guests Eva can invite.
127.2/8≈ 15.9
Since Eva wants to invite 12 guests and 12 is less than 15.9, it will be enough room for Eva's guests.
