How to find the perimeter of an acute / obtuse isosceles triangle - GRE Quantitative Reasoning

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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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