Triangles - GRE Quantitative Reasoning

Card 0 of 552

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:

$b = 4\sqrt{x}$

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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.

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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

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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

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

In the figure above, what is the value of angle x?

Answer

To find the top inner angle, recognize that a straight line contains 180o; thus we can subtract 180 – 115 = 65o. Since we are given the other interior angle of 30 degrees, we can add the two we know: 65 + 30 = 95o.

180 - 95 = 85

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Question

The three angles in a triangle measure 3_x_, 4_x_ + 10, and 8_x_ + 20. What is x?

Answer

We know the angles in a triangle must add up to 180, so we can solve for x.

3_x_ + 4_x_ + 10 + 8_x_ + 20 = 180

15_x_ + 30 = 180

15_x_ = 150

x = 10

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Question

In triangle ABC, AB=6, AC=3, and BC=4.

Quantity A Quantity B

angle C the sum of angle A and angle B

Answer

The given triangle is obtuse. Thus, angle is greater than 90 degrees. A triangle has a sum of 180 degrees, so angle + angle + angle = 180. Since angle C is greater than 90 then angle + angle must be less than 90 and it follows that Quantity A is greater.

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Question

An isosceles triangle has an angle of 110°. Which of the following angles could also be in the triangle?

Answer

An isosceles triangle always has two equal angles. As there cannot be another 110° (the triangle cannot have over 180° total), the other two angles must equal eachother. 180° - 110° = 70°. 70° represents the other two angles, so it needs to be divided in 2 to get the answer of 35°.

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Question

An isosceles triangle ABC is laid flat on its base. Given that <B, located in the lower left corner, is 84 degrees, what is the measurement of the top angle, <A?

Answer

Since the triangle is isosceles, and <A is located at the top of the triangle that is on its base, <B and <C are equivalent. Since <B is 84 degrees, <C is also. There are 180 degrees in a triangle so 180 - 84 - 84 = 12 degrees.

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Question

Triangle ABC is isosceles

x and y are positive integers

A

---

x

B

---

y

Answer

Since we are given expressions for the two congruent angles of the isosceles triangle, we can set the expressions equal to see how x relates to y. We get,

x – 3 = y – 7 --> y = x + 4

Logically, y must be the greater number if it takes x an additional 4 units to reach its value (knowing they are both positive integers).

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Question

An isosceles triangle has one obtuse angle that is $112^{\circ}$ . What is the value of one of the other angles?

Answer

We know that an isosceles triangel has two equal sides and thus two equal angles opposite those equal sides. Because there is one obtuse angle of 112 degrees we automatically know that this angle is the vertex. If you sum any triangle's interior angles, you always get 180 degrees.

180 – 112 = 68 degrees. Thus there are 68 degrees left for the two equal angles. Each angle must therefore measure 34 degrees.

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Question

What is the length of a side of an equilateral triangle if the area is $9\sqrt{3}$ ?

Answer

The area of an equilateral triangle is $\frac{s^2\sqrt{3}}{4}$ .

So let's set-up an equation to solve for .

$\frac{s^2\sqrt{3}}{4}=9\sqrt{3}$ Cross multiply.

$s^2\sqrt{3}=36\sqrt{3}$

The $\sqrt{3}$ cancels out and we get .

Then take square root on both sides and we get as the final answer.

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Question

If the height of the equilateral triangle is $4\sqrt{2}$ , then what is the length of a side of an equilateral triangle?

Answer

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is $\sqrt{3}$ and the side of equilateral triangle which is opposite is .

$\frac{4\sqrt{2}}{\sqrt{3}}=\frac{s}{2}$ Cross multiply.

$8\sqrt{2}=s\sqrt{3}$ Divide both sides by $\sqrt{3}$

$\frac{8\sqrt{2}}{\sqrt{3}}=s$ Multiply top and bottom by $\sqrt{3}$ to get rid of the radical.

$\frac{8\sqrt{6}}{3}=s$

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