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

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?

$1\frac{1}{2}$ ft

$\frac{12}{4}$ ft

$1\frac{2}{3}$ ft

$1\frac{1}{3}$ ft

Explanation

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:

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

Isos._cont._gre

Find the perimeter of the acute Isosceles triangle shown above.

$\small 132$

$\small 130$

$\small 133$

$\small 120$

$\small \small 60$

Explanation

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

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

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

$\small a=12\times5=60$

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

$\small p=120+12=132$

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

Explanation

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

Isos._cont._gre

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.

$\small 172$ inches

$\small 15\sqrt{2}$ inches

$\small 86$ inches

$\small 129$ inches

$\small 84$ inches

Explanation

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:

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

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?

$\small 7$ inches

$\small 6$ inches

$\small \frac{3}{4}$ foot

$\small \frac{1}{4}$ foot

$\small \frac{9}{12}$ foot

Explanation

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.

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.

$\small 18$ inches

$\small 19$ inches

$\small 14$ inches

$\small 12$ inches

$\small 13$ inches

Explanation

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.

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?

$\small 10$ inches

$\small 1$ foot

$\small 6$ inches

$\small \frac{4}{6}$ foot

$\small \frac{2}{6}$ foot

Explanation

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

Isos.gre_series

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?

inches

Explanation

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:

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?

$\frac{5}{4}$ yard

$\small \frac{6}{12}$ yard

$\small \frac{1}{3}$ yard

$\small \frac{1}{4}$ yard

$\small \frac{7}{12}$ yard

Explanation

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: $\small p=2a+b$

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

Isos.gre_series

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?