Plane Geometry - PSAT Math
Card 1 of 2352
The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?
The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?
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We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.
We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.
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When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?
When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?
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Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:
x2 + 64 = (x+2)2
FOIL the right side of the equation.
x2 + 64 = x2 + 4x + 4
Subtract x2 from both sides and then continue with the alegbra.
64 = 4x + 4
64 = 4(x + 1)
16 = x + 1
15 = x
Therefore, the length of the original square is 15 inches.
If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.
Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:
x2 + 64 = (x+2)2
FOIL the right side of the equation.
x2 + 64 = x2 + 4x + 4
Subtract x2 from both sides and then continue with the alegbra.
64 = 4x + 4
64 = 4(x + 1)
16 = x + 1
15 = x
Therefore, the length of the original square is 15 inches.
If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.
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If the area of a square is 25 inches squared, what is the perimeter?
If the area of a square is 25 inches squared, what is the perimeter?
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The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is 
or 
The perimeter of a square is the sum of the length of all 4 sides or 
The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is
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A circle with a radius 2 in is inscribed in a square. What is the perimeter of the square?
A circle with a radius 2 in is inscribed in a square. What is the perimeter of the square?
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To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in. The perimeter of the square = 4s = 4 * 4 = 16 in.
To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in. The perimeter of the square = 4s = 4 * 4 = 16 in.
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Square X has 3 times the area of Square Y. If the perimeter of Square Y is 24 ft, what is the area of Square X, in sq ft?
Square X has 3 times the area of Square Y. If the perimeter of Square Y is 24 ft, what is the area of Square X, in sq ft?
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Find the area of Square Y, then calculate the area of Square X.
If the perimeter of Square Y is 24, then each side is 24/4, or 6.
A = 6 * 6 = 36 sq ft, for Square Y
If Square X has 3 times the area, then 3 * 36 = 108 sq ft.
Find the area of Square Y, then calculate the area of Square X.
If the perimeter of Square Y is 24, then each side is 24/4, or 6.
A = 6 * 6 = 36 sq ft, for Square Y
If Square X has 3 times the area, then 3 * 36 = 108 sq ft.
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Note: Figure NOT drawn to scale.
Refer to the above figure.
and 
.
What percent of 
has been shaded brown ?
Note: Figure NOT drawn to scale.
Refer to the above figure.
What percent of
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and 
, so the similarity ratio of 
to 
is 10 to 7. The ratio of the areas is the square of this, or
or
Therefore, 
comprises 
of 
, and the remainder of the rectangle - the brown region - is 51% of 
.
or
Therefore,
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If it is 2:00 PM, what is the measure of the angle between the minute and hour hands of the clock?
If it is 2:00 PM, what is the measure of the angle between the minute and hour hands of the clock?
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First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2. The correct answer is 2 * 30 = 60 degrees.
First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2. The correct answer is 2 * 30 = 60 degrees.
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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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Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
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By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
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If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?
If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?
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Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.
Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.
Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.
Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.
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If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right triangle?
If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right triangle?
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Pythagorean Theorum
AB2 + BC2 = AC2
If C is 45º then A is 45º, therefore AB = BC
AB2 + BC2 = AC2
62 + 62 = AC2
2*62 = AC2
AC = √(2*62) = 6√2
Pythagorean Theorum
AB2 + BC2 = AC2
If C is 45º then A is 45º, therefore AB = BC
AB2 + BC2 = AC2
62 + 62 = AC2
2*62 = AC2
AC = √(2*62) = 6√2
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A square has an area of 
. If the side of the square is reduced by a factor of two, what is the perimeter of the new square?
A square has an area of
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The area of the given square is given by 
so the side must be 6 in. The side is reduced by a factor of two, so the new side is 3 in. The perimeter of the new square is given by 
.
The area of the given square is given by
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Note: Figure NOT drawn to scale.
In the above figure,
.
.
Give the perimeter of 
.
Note: Figure NOT drawn to scale.
In the above figure,
Give the perimeter of
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We can use the Pythagorean Theorem to find 
:
The similarity ratio of 
to 
is
so 
multiplied by the length of a side of 
is the length of the corresponding side of 
. We can subsequently multiply the perimeter of the former by 
to get that of the latter:
We can use the Pythagorean Theorem to find
The similarity ratio of
so
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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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Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?
Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?
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Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.
Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.
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A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?
A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?
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We can use the Pythagorean Theorem to solve for x.
92 + _x_2 = 152
81 + _x_2 = 225
_x_2 = 144
x = 12
We can use the Pythagorean Theorem to solve for x.
92 + _x_2 = 152
81 + _x_2 = 225
_x_2 = 144
x = 12
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A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?
A triangle has sides of length 5, 7, and x. Which of the following can NOT be a value of x?
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The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.
The sum of the lengths of any two sides of a triangle must exceed the length of the third side; therefore, 5+7 > x, which cannot happen if x = 13.
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Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?
Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?
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The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.
The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.
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If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?
If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?
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According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.
According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.
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