Plane Geometry - PSAT Math
Card 1 of 2352
ABCD is a parallelogram. BD = 5. The angles of triangle ABD are all equal. What is the perimeter of the parallelogram?
ABCD is a parallelogram. BD = 5. The angles of triangle ABD are all equal. What is the perimeter of the parallelogram?
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If all of the angles in triangle ABD are equal and line BD divides the parallelogram, then all angles in triangle BDC must be equal as well.
We now have two equilateral triangles, so all sides of the triangles will be equal.
All sides therefore equal 5.
5+5+5+5 = 20
If all of the angles in triangle ABD are equal and line BD divides the parallelogram, then all angles in triangle BDC must be equal as well.
We now have two equilateral triangles, so all sides of the triangles will be equal.
All sides therefore equal 5.
5+5+5+5 = 20
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If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?
If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?
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Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.
Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.
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The height, 
, of triangle 
in the figure is one-fourth the length of 
. In terms of h, what is the area of triangle 
?
The height, 
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If h=\frac{1}{4}$ *$\overline{PQ}$, then the length of $\overline{PQ}$ must be 4h.
Using the formula for the area of a triangle ($\frac{1}{2}$bh), with b=4h, the area of the triangle must be $2h^{2}$.
If h=\frac{1}{4}$ *$\overline{PQ}$, then the length of $\overline{PQ}$ must be 4h.
Using the formula for the area of a triangle ($\frac{1}{2}$bh), with b=4h, the area of the triangle must be $2h^{2}$.
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A triangle with two equal angles is called a(n) .
A triangle with two equal angles is called a(n) .
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An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).
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What is the measure, in degrees, of one interior angle of a regular pentagon?
What is the measure, in degrees, of one interior angle of a regular pentagon?
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The formula for the sum of the interior angles of any regular polygon is as follows:
where 
is equal to the number of sides of the regular polygon.
Therefore, the sum of the interior angles for a regular pentagon is:
To find the measure of one interior angle of a regular pentagon, simply divide by the number of sides (or number of interior angles):
The measure of one interior angle of a regular pentagon is 108 degrees.
The formula for the sum of the interior angles of any regular polygon is as follows:
where
Therefore, the sum of the interior angles for a regular pentagon is:
To find the measure of one interior angle of a regular pentagon, simply divide by the number of sides (or number of interior angles):
The measure of one interior angle of a regular pentagon is 108 degrees.
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Refer to the above figure, which shows Square 
and regular Pentagon 
.
Evaluate 
.
Refer to the above figure, which shows Square
Evaluate
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By angle addition,
is one of two acute angles of isosceles right triangle 
, so 
.
To find 
we examine 
.
is an angle of a regular pentagon and has measure 
.
Also, since, in 
, sides 
, by the Isosceles Triangle Theorem, 
.
Since the angles of a triangle must total 
in measure,
By angle addition,
To find
Also, since, in
Since the angles of a triangle must total
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Refer to the above figure.
Which of the following segments is a diagonal of Pentagon 
?
Refer to the above figure.
Which of the following segments is a diagonal of Pentagon
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A diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon. Of the five choices, only 
fits this description.
A diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon. Of the five choices, only
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Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?
Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?
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A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?
A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?
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We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)
We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)
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If the area Rectangle A is 
larger than Rectangle B and the sides of Rectangle A are 
and 
, what is the area of Rectangle B?
If the area Rectangle A is
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The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?
The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?
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Glass Area per Window = 8 ft x 6 ft = 48 ft2
Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows
Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2
Glass Area per Window = 8 ft x 6 ft = 48 ft2
Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows
Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2
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The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?
The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?
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The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.
Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.
10 feet = 10 x 12 inches = 120 inches
12 feet = 12 x 12 inches = 144 inches
This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.
Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.
Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.
If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.
The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.
The total cost = 1080 x 2.50 dollars = 2700 dollars.
The answer is $2700.
The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.
Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.
10 feet = 10 x 12 inches = 120 inches
12 feet = 12 x 12 inches = 144 inches
This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.
Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.
Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.
If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.
The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.
The total cost = 1080 x 2.50 dollars = 2700 dollars.
The answer is $2700.
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Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?
Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?
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Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.
Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.
We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.
2_l_ + 2_w_ = 4_s_
If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:
Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.
Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.
We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.
2_l_ + 2_w_ = 4_s_
If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:
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A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the area of the rectangle?
A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the area of the rectangle?
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Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.
A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_
Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.
A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_
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Two circles of a radius of 
each sit inside a square with a side length of 
. If the circles do not overlap, what is the area outside of the circles, but within the square?
Two circles of a radius of
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The area of a square = $side^{2}$
The area of a circle is pi $r^{2}$
Area = Area of Square - 2(Area of Circle) =
The area of a square = $side^{2}$
The area of a circle is pi $r^{2}$
Area = Area of Square - 2(Area of Circle) =
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George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 ft^{2} and costs $40. One quart of paint covers 100 ft^{2} and costs $15. How much money will he spend on the blue paint?
George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 ft^{2} and costs $40. One quart of paint covers 100 ft^{2} and costs $15. How much money will he spend on the blue paint?
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The area of the walls is given by 
One gallon of paint covers 400 ft^{2} and the remaining 140 ft^{2} would be covered by two quarts.
So one gallon and two quarts of paint would cost 
The area of the walls is given by
One gallon of paint covers 400 ft^{2} and the remaining 140 ft^{2} would be covered by two quarts.
So one gallon and two quarts of paint would cost
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Daisy gets new carpet for her rectangluar room. Her floor is 21 ft times 24 ft. The carpet sells for $5 per square yard. How much did she spend on her carpet?
Daisy gets new carpet for her rectangluar room. Her floor is 21 ft times 24 ft. The carpet sells for $5 per square yard. How much did she spend on her carpet?
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Since 3 ft=1 yd the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost 5cdot 56=$280.
Since 3 ft=1 yd the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost 5cdot 56=$280.
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The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?
The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?
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For a rectangle, P=2w+2l and A=lw where w is the width and l is the length.
Let x=width and 2x+5=length.
So the equation to solve becomes 40=2x+2(2x+5) or 40=6x+10.
Thus x=5 ft and 2x+5=15 ft, so the area is 75 $ft^{2}$.
For a rectangle, P=2w+2l and A=lw where w is the width and l is the length.
Let x=width and 2x+5=length.
So the equation to solve becomes 40=2x+2(2x+5) or 40=6x+10.
Thus x=5 ft and 2x+5=15 ft, so the area is 75 $ft^{2}$.
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Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) eight feet wide throughout. What is the area of that dirt path?
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) eight feet wide throughout. What is the area of that dirt path?
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The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is
square feet.
The inner rectangle has length and width 
feet and 
feet, respectively, so its area is
square feet.
The area of the path is the difference of the two:
square feet.
The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is
The inner rectangle has length and width
The area of the path is the difference of the two:
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Note: Figure NOT drawn to scale
What percent of Rectangle 
is pink?
Note: Figure NOT drawn to scale
What percent of Rectangle
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The pink region is Rectangle 
. Its length and width are
so its area is the product of these, or
.
The length and width of Rectangle 
are
so its area is the product of these, or
.
We want to know what percent 117 is of 240, which can be answered as follows:
The pink region is Rectangle
so its area is the product of these, or
The length and width of Rectangle
so its area is the product of these, or
We want to know what percent 117 is of 240, which can be answered as follows:
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