Questions on Geometry: Parallelograms answered by real tutors!

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Question 194310:
Answer by jim_thompson5910(28476) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

part i)
If you look at angles HBD, BGI, and GCA, you'll see that these angles form a triangle. Remember, the sum of all the angles in a triangle is ALWAYS 180 degrees.

So < HBD + < BGI + < GCA = 180

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part ii)

Similar to part i), the segments BG, GC, and CB form a triangle also. The union of these three segments will simply result in triangle BGC


So


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iii)
The angle GCB and the segment BH only have the points B and G in common

So the answer is simply the two points B and G.


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iv)
The angle ABG and the segment BD only have the point B in common

So the answer is simply the point B.

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# 2
a)
Supplementary angles are angles that add up to 180 degrees. In a visual sense, if you place the angles right next to one another (where the vertices meet and one sides touch), then the two angles will form a straight line.

So algebraically, this means that if "x" and "y" are two unknown angles, then x%2By=180

However, we do know that one angle is 34 degrees. So let's plug this into "y" (it doesn't matter which variable we choose)

x%2B34=180


Now subtract 34 from both sides to get x=180-34 and combine like terms to get x=146

So the supplementary angle to 34 degrees is 146 degrees.

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b)

Let x and y be the angle measurements for angles ABC and PQR respectively


Since "< ABC and < PQR are supplementary angles", this means that x%2By=180 (see above). Also, because "< ABC is eleven times as large as < PQR", we can say that x=11y.


x%2By=180 Start with the first equation


11y%2By=180 Plug in x=11y


12y=180 Combine like terms on the left side.


y=%28180%29%2F%2812%29 Divide both sides by 12 to isolate y.


y=15 Reduce. So the first angle ABC is 15 degrees.

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x=11y Go back to the second equation.


x=11%2815%29 Plug in y=15


x=165 Multiply. So the second angle PQR is 165 degrees.


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# 3


Looking at angle 3, we can see that is a vertical angle to the angle labeled with 29 degrees. Since vertical angles are equal, this means that

angle 3 = 29 degrees


Now recall that if two parallel lines are cut with a transversal lines, then alternate interior angles are equal to each other. Because angles 3 and 4 are alternate interior angles, this means that

angle 3 = angle 4 = 29 degrees

Since angles 4 and 7 are vertical angles, this means that

angle 4 = angle 7 = 29 degrees


So the following angles are 29 degrees: angles 3, 4, and 7

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Now since the top angle labeled 29 degrees and angle 1 form the line "l", this means that

29 + angle 1 = 180 degrees

Solving for angle 1, we get

angle 1 = 151


Angles 1 and 2 are vertical angles. So using the same logic, this means

angle 1 = angle 2 = 151 degrees


Angles 2 and 5 are another pair of alternate interior angles, so

angle 2 = angle 5 = 151 degrees


Finally, because angles 5 and 6 are vertical angles, this tells us that

angle 5 = angle 6 = 151 degrees


So the angles 1, 2, 5, and 6 all have measures of 151 degrees.


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# 4

First, we need to draw a picture to see what's going on:



Note: the longest side of ANY triangle is the hypotenuse.


So to find the area and perimeter, we need to find the length of the unknown side.


To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is a%5E2%2Bb%5E2=c%5E2 where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are x and 7 this means that a=x and b=7


Also, since the hypotenuse is 25, this means that c=25.


a%5E2%2Bb%5E2=c%5E2 Start with the Pythagorean theorem.


x%5E2%2B7%5E2=25%5E2 Plug in a=x, b=7, c=25


x%5E2%2B49=25%5E2 Square 7 to get 49.


x%5E2%2B49=625 Square 25 to get 625.


x%5E2=625-49 Subtract 49 from both sides.


x%5E2=576 Combine like terms.


x=sqrt%28576%29 Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


x=24 Simplify the square root.


So the unknown side is 24 cm long.

Now let's find the perimeter. The perimeter of any triangle is

Perimeter = Side 1 + Side 2 + Side 3

So
Perimeter = 7 + 24 + 25

Perimeter = 56

which means that the perimeter is 56 cm

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Now onto the area. The area of any triangle is

A=%28bh%29%2F2


A=%28bh%29%2F2 Start with the given formula


A=%2824%2A7%29%2F2 Plug in b=24 and h=7


A=168%2F2 Multiply


A=84 Reduce


So the area of the triangle is 84 square cm.


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# 5

First, recall (or look up) that there are 12 inches in a foot. So we have the equation

12 inches = 1 foot


So to convert from inches to feet, simply divide by 12. So 42%2F12=3.5 which means that

42 inches = 3.5 feet

So there are 42 inches in 3 and a half feet.


Since the diameter is 3.5 feet, this means that half of this figure is the radius which is 3.5%2F2=1.75


So the radius is r=1.75



A=pi%2Ar%5E2 Start with the area of a circle formula


A=pi%2A%281.75%29%5E2 Plug in r=1.75


A=3.0625pi Square 1.75 to get 3.0625 and rearrange the terms.


So the exact area is 3.0625pi square feet.


So the answer is C)


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# 6

The volume of a rectangular prism/box is:

Volume = Area of base * Height

So this means that the formula is

V=Ah where "A" is the area of the base.


V=Ah Start with the given formula.


V=1.2%2A2 Plug in A=1.2 and h=2


V=2.4 Multiply.


So the volume is 2.4 cubic meters which makes the answer choice B)

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# 7

Since the diameter is d=6, this means that the radius is r=d%2F2=6%2F2=3 or simply r=3

Recall, the formula for the volume of a cone is


V=%281%2F3%29%2Api%2Ar%5E2%2Ah


V=%281%2F3%29%2Api%2Ar%5E2%2Ah Start with the given formula


V=%281%2F3%29%2Api%2A3%5E2%2A10 Plug in r=3 and h=10


V=%281%2F3%29%2Api%2A9%2A10 Square 3 to get 9.


V=%2890%2F3%29%2Api Multiply and rearrange the terms.


V=30pi Reduce


V=94.248 Use 3.14 for pi and multiply


So the volume of the tank is about 94.247 cubic feet.


Since 1 cubic foot = 7.480519 gallons


this means that we simply need to multiply 94.247 by 7.480519 to get

94.247*7.480519=705.0164

Now round to the nearest hundredth to get 705.02


So the tank can hold about 705.02 gallons.


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# 8

The volume of a pyramid with a square or rectangular base is

V=%28L%2AW%2AH%29%2F3


V=%28L%2AW%2AH%29%2F3 Start with the given formula


V=%2811%2A11%2A12%29%2F3 Plug in L=11, W=11, and H=12


V=%281452%29%2F3 Multiply


V=484 Reduce


So the volume is 484 cubic meters.


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# 9

Because the diameter is 12 feet, this means that the radius is 6 feet (since the radius is half the diameter).

Remember, the formula for the volume of a cylinder is:


V=pi%2Ar%5E2%2Ah


V=pi%2Ar%5E2%2Ah Start with the given formula


V=pi%2A6%5E2%2A10 Plug in r=6 and h=10


V=pi%2A36%2A10 Square 6 to get 36


V=360pi Multiply and rearrange the terms.


So the exact volume is 360pi


If we replace pi with 3.14159, we get:

360%283.14159%29=1130.9724

Now round to the nearest hundredth to get 1130.97

So the approximate volume is 1130.97


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# 10

a)

Area of a trapezoid:

A=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2 where b%5B1%5D and b%5B2%5D are the parallel bases.


A=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2 Start with the given formula


A=%286%2814%2B21%29%29%2F2 Plug in h=6 b%5B1%5D=14 and b%5B2%5D=21


A=%286%2835%29%29%2F2 Add


A=210%2F2 Multiply


A=105 Reduce


So the area of the trapezoid is 115 square inches.

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b)

One of the many properties of a parallelogram is that opposite parallel sides are equal. So the side parallel to the side length 9 inches is also 9 inches. Similarly, the side opposite the length 5 inches is also 5 inches.

So we have the sides are of 9, 5, 9, and 5 inches

Add these lengths up to get 9+5+9+5=14+14=28

So the perimeter of the parallelogram is 28 inches.


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# 11

First, let's find the area of the upper rectangle. Simply multiply the length 11 yd by the width 5 yd to get: 11*5=55. So the area of the upper rectangle is 55 square yards.


Now let's find the area of the triangle. Looking at the figure, we see that the base is 4 yards and the height is 5 yards. Now multiply these two dimensions and divide by 2 to get: %284%2A5%29%2F2=20%2F2=10. So the area of the triangle is 10 square yards.


Finally, simply square the base of the lower square to get 4%5E2=4%2A4=16


So the total area is: 55+10+16=81 square yards which means that 81 square yards of carpeting will be needed.


Since "Carpeting will cost $8.00 a square yard", this means that we simply multiply the total area by the cost per square yard. So...

81*8=648

which means that the total cost will be $648


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# 12

Because the two triangles are congruent, this means that their corresponding parts are equal. So their two longer legs are equal which means that AB = DE = 24 cm

This means that AB = 24 cm

Also, their two hypotenuses are equal. So AC = DF = 25 cm which means that DF = 25 cm

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Also, congruent triangles have equal corresponding angles. Since angle BAC is the angle between the longer leg and the hypotenuse for the first triangle and angle EFD is the same for the second triangle, this makes

angle BAC = angle EFD = 16 degrees


So angle BAC = 16 degrees


The same applies to angles ACB and EFD. These angles are also corresponding angles. So this means that

angle ACB = angle EFD = 74 degrees

which means that angle EFD = 74 degrees


Now from before, it was stated that the sum of all angles in a triangle add to 180 degrees. Since we know two angles of the first triangle, this means that we can find the remaining angle by first adding the two angles to get: 74+16=90

Now subtract this result from 180 to get 180 - 90 = 90

So the remaining angle ABC is 90 degrees which means angle ABC = 90

Because angles ABC and DEF are corresponding angles, this tells us that angle ABC = angle DEF = 90


So angle DEF = 90


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# 13

Start with five randomly drawn points. These are the vertices




Now simply connect every vertex with a line. Make sure that EVERY point has two lines connecting to it (so a bridge doesn't form). In addition to connecting every vertex, draw a line from any one vertex to itself to form a loop (in green):