Acute / Obtuse Isosceles Triangles - GRE Quantitative Reasoning

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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

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x

B

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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 perimeter of an isosceles triangle given that the sides 5 units long and half of the base measures to 4 units?

Answer

The base of the triangle is 4 + 4 = 8 so the total perimeter is 5 + 5 + 8 = 18.

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Question

An obtuse Isosceles triangle has two sides with length and one side length . The length of side $\frac{3}{4}$ ft. If the length of half the length of side , what is the perimeter of the triangle?

Answer

By definition, an Isosceles triangle must have two equivalent side lengths. Since we are told that $b=\frac{3}{4}$ ft and that the sides with length are half the length of side , find the length of by: $\frac{3}{4}=\frac{6}{8}$ and half of $\frac{6}{8}=\frac{3}{8}$ . Thus, both of sides with length must equal $\frac{3}{8}$ ft.

Now, apply the formula: .

$p=\frac{3}{8}+\frac{3}{8}+\frac{6}{8}=\frac{12}{8}$

Then, simplify the fraction/convert to mixed number fraction:

$\frac{12}{8}=\frac{3}{2}=1\frac{1}{2}$

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Question

A triangle has two sides with length and one side length . The length of side $\frac{1}{4}$ yard. If the length of the length of side , what is the perimeter of the triangle?

Answer

The first step to solving this problem is that we must find the length of length $\small a.$ Since, $\small a$ is $\small 2$ 4 the length of side $\small b$ , use the following steps:

$\small \small b=\frac{1}{4}$
$\small a=\frac{1}{4}\times2=\frac{1}{2}$

Now, apply the formula: $\small p=2a+b$

$\small p=\frac{2}{4}+\frac{2}{4}+\frac{1}{4}=\frac{5}{4}$

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ $\small \frac{3}{9}$ ft. If the length of $\small b=$ half the length of side $\small a$ , what is the perimeter of the triangle?

Answer

This Isosceles triangle has two sides with a length of $\small \frac{3}{9}$ foot and one side length that is half of the length of the two equivalent sides.

To find the missing side, double the value of side $\small a$ 's denominator:

$\small \frac{3}{9}=\frac{1}{3}$ . Thus, half of $\small \frac{1}{3}=\frac{1}{6}$ .

Therefore, this triangle has two sides with lengths of $\small \frac{1}{3}$ and one side length of $\small \frac{1}{6}$ .

To find the perimeter, apply the formula:

$\small p=2a+b$

$\small p=\frac{1}{3}+\frac{1}{3}+\frac{1}{6}$

$\small p=\frac{2}{6}+\frac{2}{6}+\frac{1}{6}=\frac{5}{6}=\frac{10}{12}$ foot $\small = 10$ inches

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ $\small 13$ . If the length of $\small b=$ half the length of side $\small a$ , what is the perimeter of the triangle?

Answer

To solve this problem apply the formula: $\small p=2a+b$ .

However, first calculate the length of the missing side by: $\small 13\div2=6.5$ .

Thus, the solution is:

$\small p=13+13+6.5=32.5$

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Question

Find the perimeter of the acute Isosceles triangle shown above.

Answer

To solve this problem apply the formula: $\small p=2a+b$ .

However, first calculate the length of the missing side by:

$\small p=2(a)+12$

$\small a=12\times5=60$

$\small p=2(60)+12$

$\small p=120+12=132$

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Question

Find the perimeter of the acute Isosceles triangle shown above.

Answer

In order to solve this problem, first find the length of the missing sides. Then apply the formula: $\small p=2a+b$

Each of the missing sides equal:

$\small a=\frac{7}{3}\times4=\frac{28}{3}$

Then apply the perimeter formula:
$\small p=\frac{28}{3}+\frac{28}{3}+\frac{7}{3}=\frac{63}{3}=21$

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ inches. If the length of $\small b=$ $\sqrt[3]{a}$ , what is the perimeter of the triangle?

Answer

In order to solve this problem, first find the length of the missing sides. Then apply the formula: $\small p=2a+b$

The missing side equals:

$b=\sqrt[3]{a}=\sqrt[3]{8}=2$

Then plug each side length into the perimeter formula:

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ $\sqrt{144}$ inches. If the length of $\small b=a\times \frac{1}{2}$ , what is the perimeter of the triangle?

Answer

In order to solve this problem, first find the length of the missing sides. Then apply the formula: $\small p=2a+b$

The missing side equals:

$\small a=\sqrt{144}=12$

$\small b=12\times \frac{1}{2}=\frac{12}{2}=6$

Then, apply the perimeter formula by plugging in the side values:

$\small p=2(12)+6=24+6=30$

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Question

The obtuse Isosceles triangle shown above has two sides with length and one side length . The length of side $\small a=$ $\small 43$ inches. Side length $\small b=2a$ . Find the perimeter of the triangle.

Answer

To find the perimeter of this triangle, apply the perimeter formula:

$\small p=2a+b$

Since, $\small a=43$ , and $\small b=2a$ then $\small b$ must have a value of: $\small 2\times43=86$

This triangle has two side lengths of $\small 43$ inches, and one side length of $\small 86$ inches.

Thus, the solution is:

$\small p=2(43)+86=172$

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ $\small \small \frac{3}{12}$ ft. If the length of $\small b=\small a\times \frac{1}{3}$ , what is the perimeter of the triangle?

Answer

To solve this problem apply the formula: $\small p=2a+b$ .

However, first calculate the length of the missing side by:

$\small \frac{3}{12}+\frac{3}{12}+(\frac{3}{12}\times\frac{1}{3})$ , Note that $\small \frac{3}{36}=\frac{1}{12}$

$\small \small \frac{6}{12}+\frac{3}{36}=\frac{6}{12}+\frac{1}{12}=\frac{7}{12}$ .

Since, it takes $\small 12$ inches to make one foot, the perimeter is equal to $\small 7$ inches.

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Question

An acute Isosceles triangle has two sides with length and one side length . The length of side $\small a=$ $\small 8$ inches. The length of side $\small b=$ $\small a\times \frac{1}{4}$ . Find the perimeter of the triangle.

Answer

To find the perimeter of this triangle, apply the formula:

$\small p=2a+b$

$\small p=2(8)+(8\times \frac{1}{4})$

$\small p=16+\frac{8}{4}$

$\small p=16+2=18$

Note: Since this is an acute Isosceles triangle, the length of the base must be smaller than the length of both of the equivalent sides.

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