Right Triangles - GRE Quantitative Reasoning

Card 0 of 216

Question

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Answer

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

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Question

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Answer

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

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Question

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

Answer

The ratio of the sides of a 30-60-90 triangle is $\dpi{100} \small x:x{\sqrt{3}}:2x$ , with the hypotenuse being $\dpi{100} \small 2x$ . Thus, the perimeter of this triangle would be $\dpi{100} \small x+x\sqrt{3}+2x=3x\sqrt{3}$ . Since the triangle depicted in this problem has a perimeter of $\dpi{100} \small 3\sqrt{3}$ , $\dpi{100} \small x$ must equal 1, which would make the hypotenuse equal to 2.

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Question

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Answer

Use the Pythagorean theorem, $a^2 + b^2 = c^2$ , with $\dpi{100} \small a=x-4$ and $\dpi{100} \small c=x+4$ , and solve for $\dpi{100} \small b$ .

$(x-4)^2 + b^2 = (x+4)^2$

Rearrange to isolate $\dpi{100} \small b^{2}$ :

$b^2 = (x+4)^2 - (x-4)^2$

$b^2 = (x+4)(x+4) - (x-4)(x-4)$

Use FOIL to multiply out:

$b^2 = (x^2 + 8x + 16) - (x^2 - 8x + 16)$

Distribute the minus sign to rewrite without parentheses:

$b^2 = x^2 + 8x + 16 - x^2 + 8x - 16$

Combine like terms:

$b^2 = 16x$

Take the square root of both sides:

$b = 4\sqrt{x}$

Compare your answer with the correct one above

Question

Given the following triangle, what is the length of the unknown side?

Answer

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

Compare your answer with the correct one above

Question

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Answer

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Compare your answer with the correct one above

Question

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Answer

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

Compare your answer with the correct one above

Question

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

Answer

The ratio of the sides of a 30-60-90 triangle is $\dpi{100} \small x:x{\sqrt{3}}:2x$ , with the hypotenuse being $\dpi{100} \small 2x$ . Thus, the perimeter of this triangle would be $\dpi{100} \small x+x\sqrt{3}+2x=3x\sqrt{3}$ . Since the triangle depicted in this problem has a perimeter of $\dpi{100} \small 3\sqrt{3}$ , $\dpi{100} \small x$ must equal 1, which would make the hypotenuse equal to 2.

Compare your answer with the correct one above

Question

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Answer

Use the Pythagorean theorem, $a^2 + b^2 = c^2$ , with $\dpi{100} \small a=x-4$ and $\dpi{100} \small c=x+4$ , and solve for $\dpi{100} \small b$ .

$(x-4)^2 + b^2 = (x+4)^2$

Rearrange to isolate $\dpi{100} \small b^{2}$ :

$b^2 = (x+4)^2 - (x-4)^2$

$b^2 = (x+4)(x+4) - (x-4)(x-4)$

Use FOIL to multiply out:

$b^2 = (x^2 + 8x + 16) - (x^2 - 8x + 16)$

Distribute the minus sign to rewrite without parentheses:

$b^2 = x^2 + 8x + 16 - x^2 + 8x - 16$

Combine like terms:

$b^2 = 16x$

Take the square root of both sides:

$b = 4\sqrt{x}$

Compare your answer with the correct one above

Question

Given the following triangle, what is the length of the unknown side?

Answer

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

Compare your answer with the correct one above

Question

An equilateral triangle has a side length of 4. What is its height?

Answer

If an equilateral triangle is divided in 2, it forms two 30-60-90 triangles. Therefore, the side of the equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle. The side lengths of a 30-60-90 triangle adhere to the ratio x: x√3 :2x. since we know the hypothesis is 4, we also know that the base is 2 and the height is 2√3.

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Question

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Answer

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Compare your answer with the correct one above

Question

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Answer

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

Compare your answer with the correct one above

Question

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

Answer

The ratio of the sides of a 30-60-90 triangle is $\dpi{100} \small x:x{\sqrt{3}}:2x$ , with the hypotenuse being $\dpi{100} \small 2x$ . Thus, the perimeter of this triangle would be $\dpi{100} \small x+x\sqrt{3}+2x=3x\sqrt{3}$ . Since the triangle depicted in this problem has a perimeter of $\dpi{100} \small 3\sqrt{3}$ , $\dpi{100} \small x$ must equal 1, which would make the hypotenuse equal to 2.

Compare your answer with the correct one above

Question

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Answer

Use the Pythagorean theorem, $a^2 + b^2 = c^2$ , with $\dpi{100} \small a=x-4$ and $\dpi{100} \small c=x+4$ , and solve for $\dpi{100} \small b$ .

$(x-4)^2 + b^2 = (x+4)^2$

Rearrange to isolate $\dpi{100} \small b^{2}$ :

$b^2 = (x+4)^2 - (x-4)^2$

$b^2 = (x+4)(x+4) - (x-4)(x-4)$

Use FOIL to multiply out:

$b^2 = (x^2 + 8x + 16) - (x^2 - 8x + 16)$

Distribute the minus sign to rewrite without parentheses:

$b^2 = x^2 + 8x + 16 - x^2 + 8x - 16$

Combine like terms:

$b^2 = 16x$

Take the square root of both sides:

$b = 4\sqrt{x}$

Compare your answer with the correct one above

Question

Given the following triangle, what is the length of the unknown side?

Answer

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

Compare your answer with the correct one above

Question

Each of the following answer choices lists the side lengths of a different triangle. Which of these triangles does not have a right angle?

Answer

cannot be the side lengths of a right triangle. does not equal . Also, special right triangle and rules can eliminate all the other choices.

Compare your answer with the correct one above

Question

Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?

Answer

First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.

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Question

Which of the following sets of sides cannnot belong to a right triangle?

Answer

To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2. Similarly, 30-60-90 triangles have lengths x, x√3, 2x. We should also recall that 3,4,5 and 5,12,13 are special right triangles. Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.

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Question

Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?

Answer

This can be solved with the Pythagorean Theorem.

62 + 42 = _c_2

52 = _c_2

c = √52 = 2√13

Compare your answer with the correct one above

Tap the card to reveal the answer

Right Triangles - GRE Quantitative Reasoning

2 triangles are similar Triangle 1 has sides 6, 8, 10 Triangle 2 has sides 5 , 3, x find x

Draw the triangles Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship. Quantity A: Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is times the side opposite the 30 degree angle. Thus, , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is . The other two angles of the triangle are 30 degrees and 60 degrees. Quantity A: The triangle's hypotenuse length Quantity B: 2

The ratio of the sides of a 30-60-90 triangle is , with the hypotenuse being . Thus, the perimeter of this triangle would be . Since the triangle depicted in this problem has a perimeter of , must equal 1, which would make the hypotenuse equal to 2.

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Use the Pythagorean theorem, , with and , and solve for . Rearrange to isolate : Use FOIL to multiply out: Distribute the minus sign to rewrite without parentheses: Combine like terms: Take the square root of both sides:

Given the following triangle, what is the length of the unknown side?

2 triangles are similar Triangle 1 has sides 6, 8, 10 Triangle 2 has sides 5 , 3, x find x

Draw the triangles Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship. Quantity A: Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is times the side opposite the 30 degree angle. Thus, , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is . The other two angles of the triangle are 30 degrees and 60 degrees. Quantity A: The triangle's hypotenuse length Quantity B: 2

The ratio of the sides of a 30-60-90 triangle is , with the hypotenuse being . Thus, the perimeter of this triangle would be . Since the triangle depicted in this problem has a perimeter of , must equal 1, which would make the hypotenuse equal to 2.

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Use the Pythagorean theorem, , with and , and solve for . Rearrange to isolate : Use FOIL to multiply out: Distribute the minus sign to rewrite without parentheses: Combine like terms: Take the square root of both sides:

Given the following triangle, what is the length of the unknown side?

An equilateral triangle has a side length of 4. What is its height?

2 triangles are similar Triangle 1 has sides 6, 8, 10 Triangle 2 has sides 5 , 3, x find x

Draw the triangles Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship. Quantity A: Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is times the side opposite the 30 degree angle. Thus, , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is . The other two angles of the triangle are 30 degrees and 60 degrees. Quantity A: The triangle's hypotenuse length Quantity B: 2

The ratio of the sides of a 30-60-90 triangle is , with the hypotenuse being . Thus, the perimeter of this triangle would be . Since the triangle depicted in this problem has a perimeter of , must equal 1, which would make the hypotenuse equal to 2.

If the shortest side of a right triangle has length and its hypotenuse has length , what is the length of the remaining side?

Use the Pythagorean theorem, , with and , and solve for . Rearrange to isolate : Use FOIL to multiply out: Distribute the minus sign to rewrite without parentheses: Combine like terms: Take the square root of both sides:

Given the following triangle, what is the length of the unknown side?

Each of the following answer choices lists the side lengths of a different triangle. Which of these triangles does not have a right angle?

cannot be the side lengths of a right triangle. does not equal . Also, special right triangle and rules can eliminate all the other choices.

Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?

Which of the following sets of sides cannnot belong to a right triangle?

Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?

This can be solved with the Pythagorean Theorem. 62 + 42 = _c_2 52 = _c_2 c = √52 = 2√13

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship.

Quantity A: $\dpi{100} \small x$

Quantity B: 7.5

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is $\dpi{100} \small \sqrt{3}$ times the side opposite the 30 degree angle. Thus, $\dpi{100} \small 5\sqrt{3}$ , which is about 8.66. This is larger than 7.5.

A right triangle's perimeter is $\dpi{100} \small 3+\sqrt{3}$ . The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

This can be solved with the Pythagorean Theorem.

62 + 42 = _c_2

52 = _c_2

c = √52 = 2√13