In a regular octagon, AB is a diagonal and CD joins the midpoints of two opposite sides. The side length of the octagon is 4 cm. To the nearest tenth of a cm. find a)AB and b)CD. I'm stumped.
form a small right-angled isosceles triangle by extending the horizontal side and the vertical side at the top right of your figure.
this triangle has a hypotenuse of 4 cm
let each of its sides be x
then x^2 + x^2 = 16
x = √8 or 2√2
I dres CD horizontally, so clearly CD = 2√2 + 4 + 2√2 = 4 + 4√2
I then constructed a large right-angled triangle inside the octogon with AB as the hypotenuse.
Then the long side must be the length of CD or (4+4√2) and the shorter side must be 4
AB^2 = (4+4√2)^2 + 4^2
Don't know if you need your answer in radical form or as a decimal, I will let you decide.
Thanks so much! If I may pick at your brain, how did you know to draw the isosceles triangle at the top and the large triangle inside the octogan to solve this problem.I didn't even consider this.
Cathy, no easy answer to your last question.
I guess teaching math for 35 years might have had something to do with it, lol
Generally, if I have geometry problems like this, I start by drawing all kinds of lines. Sometimes I get results, sometimes it leads to dead ends.
Thanks again. I really appreciate it.
To solve this problem, we can use the properties of regular octagons and geometric relationships between the given lines.
Let's start by drawing a regular octagon and labeling the given information:
A ------- B
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D ------- C
Now, let's find the lengths of AB and CD step by step.
a) Length of AB:
In a regular octagon, every diagonal divides the octagon into two congruent trapezoids. Since AB is a diagonal, we can draw two congruent trapezoids by connecting AC and BC.
In this case, the trapezoids are ADCB and ADBC.
Since ABCD is a regular octagon, all the sides are equal.
Let's consider triangle ADC. It is an isosceles triangle because AD = DC (as D is the midpoint).
To find the length of AB, we need to find the length of AD first.
Using the Pythagorean theorem, we can find AD:
AD^2 = AC^2 - CD^2
The length of AC can be found by considering triangle ABC:
AC = √2 * AB
Similarly, the length of CD can be found by considering triangle ADC:
CD = √2 * AD
Let's substitute these values into the equation for AD:
AD^2 = (√2 * AB)^2 - (√2 * AD)^2
AD^2 = 2AB^2 - 2AD^2
Rearranging the equation,
3AD^2 = 2AB^2
Since AD = CD (as D is midpoint),
3CD^2 = 2AB^2
Now, we can substitute the given values:
3(4/2)^2 = 2AB^2
3(2)^2 = 2AB^2
12 = 2AB^2
6 = AB^2
AB = √6 ≈ 2.45 cm
Therefore, to the nearest tenth of a cm, AB ≈ 2.5 cm.
b) Length of CD:
We already found that CD = AD.
Using the Pythagorean theorem again, we can find AD:
AD^2 = AC^2 - CD^2
Let's substitute the given values:
AD^2 = (√2 * AB)^2 - CD^2
AD^2 = 2AB^2 - CD^2
Since AD = CD,
CD^2 = 2AB^2 - CD^2
2CD^2 = 2AB^2
CD^2 = AB^2
Substituting the value of AB:
CD^2 = (2.5)^2
CD^2 = 6.25
CD = √6.25 ≈ 2.5 cm
Therefore, to the nearest tenth of a cm, CD ≈ 2.5 cm.