Plane Geometry - PSAT Math
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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.
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.
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.
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
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.
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
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
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.
A square has an area of 36 cm2. A circle is inscribed and cut out. What is the area of the remaining shape? Use 3.14 to approximate π.
A square has an area of 36 cm2. A circle is inscribed and cut out. What is the area of the remaining shape? Use 3.14 to approximate π.
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We need to find the area of both the square and the circle and then subtract the two. Inscribed means draw within a figure so as to touch in as many places as possible. So the circle is drawn inside the square. The opposite is circumscribed, meaning drawn outside.
Asquare = s2 = 36 cm2 so the side is 6 cm
6 cm is also the diameter of the circle and thus the radius is 3 cm
A circle = πr2 = 3.14 * 32 = 28.28 cm2
The resulting difference is 7.74 cm2
We need to find the area of both the square and the circle and then subtract the two. Inscribed means draw within a figure so as to touch in as many places as possible. So the circle is drawn inside the square. The opposite is circumscribed, meaning drawn outside.
Asquare = s2 = 36 cm2 so the side is 6 cm
6 cm is also the diameter of the circle and thus the radius is 3 cm
A circle = πr2 = 3.14 * 32 = 28.28 cm2
The resulting difference is 7.74 cm2
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.
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
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.
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.
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.
A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?
A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?
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In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.
The area of a square is found by multiply the lengths of 2 sides of a square by itself.
So, the square root of 3,600 comes out to 60 ft.
The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.
602 + 602 = C2
the square root of 7,200 is 84.8, which can be rounded to 85
In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.
The area of a square is found by multiply the lengths of 2 sides of a square by itself.
So, the square root of 3,600 comes out to 60 ft.
The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.
602 + 602 = C2
the square root of 7,200 is 84.8, which can be rounded to 85
Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?
Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?
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We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.
Apply the Pythagorean Theorem:
a2 + b2 = c2
25 + 64 = c2
89 = c2
c = 9.43 miles
We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.
Apply the Pythagorean Theorem:
a2 + b2 = c2
25 + 64 = c2
89 = c2
c = 9.43 miles
If the perimeter of a square is equal to twice its area, what is the length of one of its sides?
If the perimeter of a square is equal to twice its area, what is the length of one of its sides?
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Area of a square in terms of each of its sides:
Area = S x S
Perimeter of a square:
Perimeter = 4S
So if 'the perimeter of a square is equal to twice its area':
2 x Area = Perimeter
2 x \[S x S\] = \[4S\]; divide by 2:
S x S = 2S; divide by S:
S = 2
Area of a square in terms of each of its sides:
Area = S x S
Perimeter of a square:
Perimeter = 4S
So if 'the perimeter of a square is equal to twice its area':
2 x Area = Perimeter
2 x \[S x S\] = \[4S\]; divide by 2:
S x S = 2S; divide by S:
S = 2
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.
In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?
In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?
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Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.
Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.
Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.
Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.
The length of an arc, 
, of a circle is 
and the radius, 
, of the circle is 
. What is the measure in degrees of the central angle, 
, formed by the arc 
?
The length of an arc,
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The circumference of the circle is 
.
The length of the arc S is 
.
A ratio can be established:
Solving for _
_yields 90o.
Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.
The circumference of the circle is
The length of the arc S is
A ratio can be established:
Solving for _
Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.