Geometry - PSAT Math
Card 1 of 4697
Solve the equation for x and y.
x/y = 30
x + y = 5
Solve the equation for x and y.
x/y = 30
x + y = 5
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Similar problem to the one before, with x being divided by y instead of multiplied. Solve in the same manner but keep in mind the way that x/y is graphed. We end up solving for one solution. The graph below illustrates the solution,
Similar problem to the one before, with x being divided by y instead of multiplied. Solve in the same manner but keep in mind the way that x/y is graphed. We end up solving for one solution. The graph below illustrates the solution,
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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.
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Two points on line m are (3,7) and (-2, 5). Line k is perpendicular to line m. What is the slope of line k?
Two points on line m are (3,7) and (-2, 5). Line k is perpendicular to line m. What is the slope of line k?
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The slope of line m is the (y2 - y1) / (x2 - x1) = (5-7) / (-2 - 3)
= -2 / -5
= 2/5
To find the slope of a line perpendicular to a given line, we must take the negative reciprocal of the slope of the given line.
Thus the slope of line k is the negative reciprocal of 2/5 (slope of line m), which is -5/2.
The slope of line m is the (y2 - y1) / (x2 - x1) = (5-7) / (-2 - 3)
= -2 / -5
= 2/5
To find the slope of a line perpendicular to a given line, we must take the negative reciprocal of the slope of the given line.
Thus the slope of line k is the negative reciprocal of 2/5 (slope of line m), which is -5/2.
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The equation of a line is: 8x + 16y = 48
What is the slope of a line that runs perpendicular to that line?
The equation of a line is: 8x + 16y = 48
What is the slope of a line that runs perpendicular to that line?
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First, solve for the equation of the line in the form of y = mx + b so that you can determine the slope, m of the line:
8x + 16y = 48
16y = -8x + 48
y = -(8/16)x + 48/16
y = -(1/2)x + 3
Therefore the slope (or m) = -1/2
The slope of a perpendicular line is the negative inverse of the slope.
m = - (-2/1) = 2
First, solve for the equation of the line in the form of y = mx + b so that you can determine the slope, m of the line:
8x + 16y = 48
16y = -8x + 48
y = -(8/16)x + 48/16
y = -(1/2)x + 3
Therefore the slope (or m) = -1/2
The slope of a perpendicular line is the negative inverse of the slope.
m = - (-2/1) = 2
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The diameter of a circle has endpoints at points (2, 10) and (–8, –14). Which of the following points does NOT lie on the circle?
The diameter of a circle has endpoints at points (2, 10) and (–8, –14). Which of the following points does NOT lie on the circle?
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Solve the equation for x and y.
x – y = 26/17
2_x_ + 3_y_ = 2
Solve the equation for x and y.
x – y = 26/17
2_x_ + 3_y_ = 2
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Straightforward problem that presents two unknowns with two equations. The student will need to deal with the fractions correctly to get this one right. Other than the fraction the problem is solved in the exact same manner as the rest in this set. The graph below illustrates the solution.
Straightforward problem that presents two unknowns with two equations. The student will need to deal with the fractions correctly to get this one right. Other than the fraction the problem is solved in the exact same manner as the rest in this set. The graph below illustrates the solution.
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What is the equation for a line with endpoints (-1, 4) and (2, -5)?
What is the equation for a line with endpoints (-1, 4) and (2, -5)?
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First we need to find the slope. Slope (m) = (y2 - y1)/(x2 - x1). Substituting in our values (-5 - 4)/(2 - (-1)) = -9/3 = -3 so slope = -3. The formula for a line is y = mx +b. We know m = -3 so now we can pick one of the two points, substitute in the values for x and y, and find b. 4 = (-3)(-1) + b so b = 1. Our formula is thus y = -3x + 1
First we need to find the slope. Slope (m) = (y2 - y1)/(x2 - x1). Substituting in our values (-5 - 4)/(2 - (-1)) = -9/3 = -3 so slope = -3. The formula for a line is y = mx +b. We know m = -3 so now we can pick one of the two points, substitute in the values for x and y, and find b. 4 = (-3)(-1) + b so b = 1. Our formula is thus y = -3x + 1
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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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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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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 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
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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
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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
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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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