Calculating the perimeter of a right triangle - GMAT Quantitative Reasoning

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Question

What is the perimeter of a traingle?

Answer

A triangle is a right triangle. To find the perimeter, we must add up all the sides of the triangle.

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Question

What is the perimeter of a 30-60-90 triangle?

One of the sides measures 10 inches.

One of the sides measures 20 inches.

Answer

The hypotenuse of a 30-60-90 triangle measures twice its shorter leg; its longer leg measures $\small \sqrt{3}$ times its shorter leg. Given one side length alone, there is no indication which of the three sides it measures; but given two, one of which is twice the other, as is the case here (10 and 20), 10 must be the shorter leg and 20 the hypotenuse. The longer leg is therefore $10\sqrt{3}$ , and the perimeter is

$P=10+20+10\sqrt{3}=30+10\sqrt{3}$ .

Therefore, both statements together are sufficent but neither alone is sufficient.

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Question

A right triangle has legs of length feet and and $6 \frac{2}{3}$ feet. Give the perimeter of this triangle in yards.

Answer

We can use the Pythagorean Theorem to calculate the hypotenuse of the triangle by setting $a = 5, b = 6 \frac{2}{3}= \frac{20}{3}$ in this formula:

$c^{2} = a ^{2}+ b^{2}$

$c^{2} = 5 ^{2}+ \left ( \frac{20}{3} \right ) ^{2}$

$c^{2} = 25 + \frac{400}{9} = \frac{225}{9} + \frac{400}{9} = \frac{625}{9}$

$c = \sqrt{\frac{625}{9} } = \frac{25}{3} = 8 \frac{1}{3}$

The perimeter is $P = a + b + c = 5 + 6\frac{2}{3} + 8 \frac{1}{3} = 20$ feet.

Divide by 3 to convert to yards:

$20 \div 3 = 6 \frac{2}{3}$ yards

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Question

A right triangle has a hypotenuse of length 13 yards and a leg of length 5 yards. Give the perimeter of this triangle in inches.

Answer

We can use the Pythagorean Theorem to calculate the length of the second leg in yards of the triangle by setting in this formula:

$b ^{2} = c ^{2 } - a ^{2}$

$b ^{2} = 13 ^{2 } - 5 ^{2} = 169 - 25 = 144$

$b = \sqrt{ 144} = 12$

The perimeter of the triangle is yards is

yards. Multiply this by 36 to covert to inches:

$30 \times 36 = 1,080$ inches.

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Question

A right triangle has a base of 4 and a height of 3. What is the perimeter of the triangle?

Answer

We are given two sides of the right triangle, so in order to calculate the perimeter we must first find the length of the third side, the hypotenuse, using the Pythagorean theorem:

$c^2=25\rightarrow c=5$

Now that we know the length of the third side, we can add the lengths of the three sides to calculate the perimeter of the right triangle:

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Question

$\Delta BCD$ is a right triangle and and . What is half the circle's circumference added to the triangle's perimeter?

Answer

As you see that a right triangle as sides 3 and 4, you should always remember that, this right triangle is a Pythagorean Triple, in other words, its sides will be in the ratio where is a constant, this will save you a lot of time. Here we can say that the hypotenuse will be 5. Therefore, the circumference will be $5\pi$ . To get the final answer, we should just divide the circumference by 2 and add the perimeter, being or 12.

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