Geometry - Common Core: 8th Grade Math

Card 0 of 20

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, notice that the black angle rotates $90^\circ$ counterclockwise, or left around the y-axis. The vertical, base, line of the angle goes from being vertical to horizontal; thus the transformation is a rotation.

The transformation can't be a reflection over the x-axis because the orange angle didn't flip over the x-axis.

The transformation can't be a translation because the angle changes direction, which does not happened when you simply move or slide an angle or image.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, notice that the black angle rotates $1800^\circ$ counterclockwise, or left around the y-axis. The vertical, base, line of the angle goes from being the base, to the top; thus the transformation is a rotation.

The transformation can't be a reflection over the x-axis because the orange angle didn't flip over the x-axis.

The transformation can't be a translation because the angle changes direction, which does not happened when you simply move or slide an angle or image.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, notice that the black angle rotates $90^\circ$ clockwise, or right around the x-axis. The vertical, base, line of the angle goes from being vertical to horizontal; thus the transformation is a rotation.

The transformation can't be a reflection over the y-axis because the orange angle didn't flip over the y-axis.

The transformation can't be a translation because the angle changes direction, which does not happened when you simply move or slide an angle or image.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, notice that the black angle rotates $1800^\circ$ clockwise, or right around the x-axis. The vertical, base, line of the angle goes from being the base, to the top; thus the transformation is a rotation.

The transformation can't be a reflection over the y-axis because the orange angle didn't flip over the y-axis.

The transformation can't be a translation because the angle changes direction, which does not happened when you simply move or slide an angle or image.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $90^\circ$ because that rotation would have caused the vertical, base, line of the angle to go from being horizontal to vertical, but the line is still horizontal. The line was not moved down, as the translation is described in the answer choice, because you can tell the angle has been flipped, the straight, base line of the angle is now the top line of the angle; thus, the correct answer is a reflection over the x-axis.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $90^\circ$ because that rotation would have caused the vertical, base, line of the angle to go from being horizontal to vertical, but the line is still horizontal. The line was not moved to the left, as the translation is described in the answer choice, because you can tell the angle has been flipped, the opening of the angle is facing the opposite direction; thus, the correct answer is a reflection over the y-axis.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $90^\circ$ because that rotation would have caused the vertical, base, line of the angle to go from being horizontal to vertical, but the line is still horizontal. The line was not reflected over the y-axis because the angle was not flipped and the opening of the angle is not facing the opposite direction; thus, the correct answer is a translation down and at diagonal.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $90^\circ$ because that rotation would have caused the vertical, base, line of the angle to go from being horizontal to vertical, but the line is still horizontal. The line was not reflected over the y-axis because the angle was not flipped and the opening of the angle is not facing the opposite direction; thus, the correct answer is a translation up and to the left.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $90^\circ$ because that rotation would have caused the vertical, base, line of the angle to go from being horizontal to vertical, but the line is still horizontal. The line was not reflected over the x-axis because the angle was not flipped and the base of the angle is the straight line, like that of the black angle; thus, the correct answer is a translation down.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $10^\circ$ because that rotation would have caused direction of the angle to turn a bit, but the angle's direction did not change. The line was not reflected over the y-axis because the angle was not flipped and the opening of the angle is not facing the opposite direction; thus, the correct answer is a translation up and to the left.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $10^\circ$ because that rotation would have caused direction of the angle to turn a bit, but the angle's direction did not change. The line was not reflected over the y-axis because the angle was not flipped and the opening of the angle is not facing the opposite direction; thus, the correct answer is a translation to the left.

Compare your answer with the correct one above

Question

Observe the location of the black and orange angles on the provided coordinate plane and identify which of the following transformations—rotation, translation, or reflection—the black angle has undergone in order to reach the position of the orange angle. Select the answer that provides the correct transformation shown in the provided image.

Answer

First, let's define the possible transformations.

Rotation: A rotation means turning an image, shape, line, etc. around a central point.

Translation: A translation means moving or sliding an image, shape, line, etc. over a plane.

Reflection: A reflection mean flipping an image, shape, line, etc. over a central line.

In the images from the question, the line was not rotated $10^\circ$ because that rotation would have caused direction of the angle to turn a bit, but the angle's direction did not change. The line was not reflected over the y-axis because the angle was not flipped and the opening of the angle is not facing the opposite direction; thus, the correct answer is a translation up and to the right.

Compare your answer with the correct one above

Question

An 8-foot-tall tree is perpendicular to the ground and casts a 6-foot shadow. What is the distance, to the nearest foot, from the top of the tree to the end of the shadow?

Answer

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6:

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

$\dpi{100} \small a^{2}+b^{2}=c^{2}$

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10.

Compare your answer with the correct one above

Question

Triangle ABC is a right triangle. If the length of side A = 3 inches and C = 5 inches, what is the length of side B?

Answer

Using the Pythagorean Theorem, we know that $a^{2} + b^{2} = c^{2}$ .

This gives:

$3^{2} + b^{2} = 5^{2}$

$9 + b^{2} = 25$

Subtracting 9 from both sides of the equation gives:

$b^{2} = 16$

inches

Compare your answer with the correct one above

Question

Find the perimeter of the polygon.

Answer

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, , where and are the legs of the triangle and is its hypotenuse.

$\sqrt{100} = \sqrt{c^2}$

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

Compare your answer with the correct one above

Question

The base and height of a right triangle are each 1 inch. What is the hypotenuse?

Answer

You need to use the Pythagorean Theorem, which is $a^{2}+b^{^{2}} = c^{^{2}}$ .

Add the first two values and you get . Take the square root of both sides and you get $\sqrt{2}$ .

Compare your answer with the correct one above

Question

Refer to the above diagram, which depicts a right triangle. What is the value of ?

Answer

By the Pythagorean Theorem, which says . being the hypotenuse, or in this problem.

$x^{2} = 6^{2} + 9^{2}$

$x^{2} = 36 + 81$

$x^{2} = 117$

$x =\sqrt{ 117}$

Simply

$\sqrt{ 9} \cdot \sqrt{ 13}$

$3 \sqrt{ 13}$

Compare your answer with the correct one above

Question

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Answer

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

$a^{2}+b^{2}=c^{2}$

We know that the base is , so we can substitute in for . We also know that the height is , so we can substitute in for .

$7^{2}+4^{2}=c^{2}$

Next we evaluate the exponents:

$7^{2}=49$

$4^{2}=16$

Now we add them together:

Then, $65=c^{2}$ .

is not a perfect square, so we simply write the square root as $\sqrt{65}$ .

$c=\sqrt{65}$

Compare your answer with the correct one above

Question

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Answer

To solve this problem, we are going to use the Pythagorean Theorom, which states that $a^{2}+b^{2}=c^{2}$ .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

$3^{2}+9^{2}=c^{2}$

Next, we evaluate the expoenents:

$3^{2}=9$

$9^{2}=81$

$9+81=c^{2}$

Then, $90=c^{2}$ .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply $c=\sqrt{90}$ .

Compare your answer with the correct one above

Question

What is the hypotenuse of a right triangle with sides 5 and 8?

Answer

According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is $a^{2} + b^{2}=c^{2}$ . Plugging in the sides, we get $5^{2}+8^{2}=c^{2}$ . Solving for , we find that the hypotenuse is $\sqrt{89}$ :

Compare your answer with the correct one above

Tap the card to reveal the answer