Plane Geometry - Math
Card 1 of 2152
If 2 sides of the triangle are have lengths equal to 8 and 14, what is one possible length of the third side?
If 2 sides of the triangle are have lengths equal to 8 and 14, what is one possible length of the third side?
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The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.
The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.
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In 
the length of AB is 15 and the length of side AC is 5. What is the least possible integer length of side BC?
In
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Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.
Given lengths of two of the sides of the 
are 15 and 5. The length of the third side must be greater than 15-5 or 10 and less than 15+5 or 20.
The question asks what is the least possible integer length of BC, which would be 11
Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.
Given lengths of two of the sides of the
The question asks what is the least possible integer length of BC, which would be 11
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Two similiar triangles exist where the ratio of perimeters is 4:5 for the smaller to the larger triangle. If the larger triangle has sides of 6, 7, and 12 inches, what is the perimeter, in inches, of the smaller triangle?
Two similiar triangles exist where the ratio of perimeters is 4:5 for the smaller to the larger triangle. If the larger triangle has sides of 6, 7, and 12 inches, what is the perimeter, in inches, of the smaller triangle?
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The larger triangle has a perimeter of 25 inches. Therefore, using a 4:5 ratio, the smaller triangle's perimeter will be 20 inches.
The larger triangle has a perimeter of 25 inches. Therefore, using a 4:5 ratio, the smaller triangle's perimeter will be 20 inches.
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To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
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Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
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A right triangle has legs of 15m and 20m. What is the length of the hypotenuse?
A right triangle has legs of 15m and 20m. What is the length of the hypotenuse?
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The Pythagorean theorem is a2 + b2 = c2, where a and b are legs of the right triangle, and c is the hypotenuse.
(15)2 + (20)2 = c2 so c2 = 625. Take the square root to get c = 25m
The Pythagorean theorem is a2 + b2 = c2, where a and b are legs of the right triangle, and c is the hypotenuse.
(15)2 + (20)2 = c2 so c2 = 625. Take the square root to get c = 25m
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A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
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If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.
If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.
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A circle has the equation below. What is the circumference of the circle?
(x – 2)2 + (y + 3)2 = 9
A circle has the equation below. What is the circumference of the circle?
(x – 2)2 + (y + 3)2 = 9
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The radius is 3. Yielding a circumference of 
.
The radius is 3. Yielding a circumference of
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The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
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We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)
For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5
If d = √5, the circumference of our circle is πd, or π√5.
We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)
For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5
If d = √5, the circumference of our circle is πd, or π√5.
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What is the magnitude of the interior angle of a regular nonagon?
What is the magnitude of the interior angle of a regular nonagon?
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The equation to calculate the magnitude of an interior angle is 
, where 
is equal to the number of sides.
For our question, 
.
The equation to calculate the magnitude of an interior angle is
For our question,
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Length AB = 4
Length BC = 3
If a similar triangle has a hypotenuse length of 25, what are the lengths of its two legs?
Length AB = 4
Length BC = 3
If a similar triangle has a hypotenuse length of 25, what are the lengths of its two legs?
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Similar triangles are in proportion.
Use Pythagorean Theorem to solve for AC:
Pythagorean Theorem: _AB_2 + _BC_2 = _AC_2
42 + 32 = _AC_2
16 + 9 = _AC_2
25 = _AC_2
AC = 5
If the similar triangle's hypotenuse is 25, then the proportion of the sides is AC/25 or 5/25 or 1/5.
Two legs then are 5 times longer than AB or BC:
5 * (AB) = 5 * (4) = 20
5 * (BC) = 5 * (3) = 15
Similar triangles are in proportion.
Use Pythagorean Theorem to solve for AC:
Pythagorean Theorem: _AB_2 + _BC_2 = _AC_2
42 + 32 = _AC_2
16 + 9 = _AC_2
25 = _AC_2
AC = 5
If the similar triangle's hypotenuse is 25, then the proportion of the sides is AC/25 or 5/25 or 1/5.
Two legs then are 5 times longer than AB or BC:
5 * (AB) = 5 * (4) = 20
5 * (BC) = 5 * (3) = 15
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What is the interior angle measure of any regular heptagon?
What is the interior angle measure of any regular heptagon?
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To find the angle of any regular polygon you find the number of sides, 
. In this example, 
.
You then subtract 2 from the number of sides yielding 5.
Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon. 
Then to find one individual angle we divide 900 by the total number of angles, 7.
The answer is 
.
To find the angle of any regular polygon you find the number of sides,
You then subtract 2 from the number of sides yielding 5.
Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon.
Then to find one individual angle we divide 900 by the total number of angles, 7.
The answer is
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A regular polygon with 
sides has exterior angles that measure 
each. How many sides does the polygon have?
A regular polygon with
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The sum of the exterior angles of any polygon, one per vertex, is 
. As each angle measures 
, just divide 360 by 1.5 to get the number of angles.
The sum of the exterior angles of any polygon, one per vertex, is
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What is the interior angle measure of any regular nonagon?
What is the interior angle measure of any regular nonagon?
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To find the angle of any regular polygon you find the number of sides 
, which in this example is 
.
You then subtract 
from the number of sides yielding 
.
Take 
and multiply it by 
degrees to yield a total number of degrees in the regular nonagon.
Then to find one individual angle we divide 
by the total number of angles 
.
The answer is 
.
To find the angle of any regular polygon you find the number of sides
You then subtract
Take
Then to find one individual angle we divide
The answer is
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What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?
What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?
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The sum of the measures of the exterior angles of any polygon, one per vertex, is 
. In a regular polygon, all of these angles are congruent, so divide 360 by 17 to get the measure of one exterior angle:
The sum of the measures of the exterior angles of any polygon, one per vertex, is
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What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
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The sum of the measures of the exterior angles of any polygon, one per vertex, is 
. In a regular polygon, all of these angles are congruent, so divide 360 by 23 to get the measure of one exterior angle:
The sum of the measures of the exterior angles of any polygon, one per vertex, is
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What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
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The measure of each interior angle of a regular polygon with 
sides is 
. We can substitute 
to obtain the angle measure:
The measure of each interior angle of a regular polygon with
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A regular polygon has interior angles which measure 
each. How many sides does the polygon have?
A regular polygon has interior angles which measure
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The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure 
. Since 360 divided by the number of sides of a regular polygon is equal to the measure of one of its exterior angles, we are seeking 
such that
Solve for 
:
The polygon has 20 sides.
The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure
Solve for
The polygon has 20 sides.
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What is the area of a regular heptagon with an apothem of 4 and a side length of 6?
What is the area of a regular heptagon with an apothem of 4 and a side length of 6?
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What is the area of a regular heptagon with an apothem of 4 and a side length of 6?
To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
First, we must calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.
In a heptagon the number of sides is 7 and in this example the side length is 6 so 
The perimeter is 
.
Then we plug in the numbers for the apothem and perimeter into the equation yielding 
We then multiply giving us the area of 
.
What is the area of a regular heptagon with an apothem of 4 and a side length of 6?
To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
First, we must calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.
In a heptagon the number of sides is 7 and in this example the side length is 6 so
The perimeter is
Then we plug in the numbers for the apothem and perimeter into the equation yielding
We then multiply giving us the area of
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What is the area of a regular decagon with an apothem of 15 and a side length of 25?
What is the area of a regular decagon with an apothem of 15 and a side length of 25?
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To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
First, we must calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.
In a decagon the number of sides is 10 and in this example the side length is 25 so 
The perimeter is 
.
Then we plug in the numbers for the apothem and perimeter into the equation yielding 
We then multiply giving us the area of 
.
To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
First, we must calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.
In a decagon the number of sides is 10 and in this example the side length is 25 so
The perimeter is
Then we plug in the numbers for the apothem and perimeter into the equation yielding
We then multiply giving us the area of
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What is the area of a regular heptagon with an apothem of 
and a side length of 
?
What is the area of a regular heptagon with an apothem of
Tap to reveal answer
To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
We must then calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides 
.
In a heptagon the number of sides 
is 
and in this example the side length is 
so 
The perimeter is 56.
Then we plug in the numbers for the apothem and perimeter into the equation yielding 
We then multiply giving us the area of 
.
To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is
We must then calculate the perimeter using the side length.
To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides
In a heptagon the number of sides
The perimeter is 56.
Then we plug in the numbers for the apothem and perimeter into the equation yielding
We then multiply giving us the area of
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