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# geometry 2

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READ FIRST *I am so sorry for posting multiple, but my computer is really glitchy and it wouldn't let me post, giving me a small box many times, so when I finally got it to work like once or twice, I though that I might as well put it all in one post to save struggles later on.*

1. The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of ∠ABC?

2. As shown in the diagram, AB and CD are two chords of circle O. The extension of the two chords intersect outside the circle at P. If PC = CD = 5 and PA = 4, find AB.

3. Rectangle ABCD is inscribed in circle O. We know AB = 4 and BC = 6. What is the combined area of the gray regions?

Mar 2, 2021

#2
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1.

we can first start off by figuring out how many degrees are in each interior angle of a hexagon: (6-2)*180$^\circ$=720$^\circ$           $\frac{720^\circ}{6}=120^\circ$  and we know that the triangle with B and C as the two base vertices is an isoscolese, because the square and the hexagon have equal side lengths, so angle B = angle C. so we have

$\frac{180-(120-90)}/2$= angle C = $75^\circ$

and because we want to find angle ABC, we have to subtract 75$^\circ$ from 120 to get angle ABC = 120-75 = $\boxed{45^\circ}$

2.

Because of power of point, we know that PC$\cdot$CD = PA$\cdot$AB = 25 = 4AB

so AB=$\boxed{\frac{25}4}$

3.

The diagonal of rectangle ABCD is the diameter of the circle, so

4$^2$+6$^2$= CD^2 = 52

CD= 2$\sqrt{13}$

so the area of the square is:

$(\frac{2\sqrt{13}}2)^2\cdot\pi=13\pi$

so

shaded region = $\boxed{13\pi-24}$

Mar 2, 2021
edited by SparklingWater2  Mar 2, 2021
edited by SparklingWater2  Mar 2, 2021

#1
+117660
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First  one

Call the triangle  BCD

CD  = BD

Angle   CDB   =  120 - 90 = 30

So  angle CBD  = (180 - 30)   / 2  =  150 /2  = 75

So angle  ABC    = 120  -  75   =  45°

Mar 2, 2021
#6
+507
+2

thank you!

iamhappy  Mar 2, 2021
#2
+360
+3

1.

we can first start off by figuring out how many degrees are in each interior angle of a hexagon: (6-2)*180$^\circ$=720$^\circ$           $\frac{720^\circ}{6}=120^\circ$  and we know that the triangle with B and C as the two base vertices is an isoscolese, because the square and the hexagon have equal side lengths, so angle B = angle C. so we have

$\frac{180-(120-90)}/2$= angle C = $75^\circ$

and because we want to find angle ABC, we have to subtract 75$^\circ$ from 120 to get angle ABC = 120-75 = $\boxed{45^\circ}$

2.

Because of power of point, we know that PC$\cdot$CD = PA$\cdot$AB = 25 = 4AB

so AB=$\boxed{\frac{25}4}$

3.

The diagonal of rectangle ABCD is the diameter of the circle, so

4$^2$+6$^2$= CD^2 = 52

CD= 2$\sqrt{13}$

so the area of the square is:

$(\frac{2\sqrt{13}}2)^2\cdot\pi=13\pi$

so

shaded region = $\boxed{13\pi-24}$

SparklingWater2 Mar 2, 2021
edited by SparklingWater2  Mar 2, 2021
edited by SparklingWater2  Mar 2, 2021
#4
+507
+2

wow! thank you so much!

iamhappy  Mar 2, 2021
#3
+836
0

Hello iamhappy!

I'm gonna try the third one. :))

AC is the diameter of the circle, and with pythagreon theorum. 4^2+6^2 = 52, sqrt(52)

So, the raduis is sqrt(13) since it's half the diameter.

The area of the circle is 13pi (sqrt(13)^2*pi)

And the area of the white rectangle is 4*6 = 24

So the area of the grey is 13pi-24.

I hope this helped. :))

=^._.^=

Mar 2, 2021
edited by catmg  Mar 2, 2021
#5
+507
+2

thank you!

iamhappy  Mar 2, 2021