How to find the perimeter of an acute / obtuse triangle - Intermediate Geometry

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Question

An acute isosceles triangle has an area of square units and a base with length . Find the perimeter of this triangle.

Answer

To solve this problem, first work backwards using the formula:

$Area=\frac{b\cdot h}{2}$

Plugging in the given information and solving for height.

$24=\frac{6h}{2}$

$24\times2=6h$

$h=\frac{48}{6}=8$

Now that you've found the height of the triangle, use the Pythagorean Theorem to find the length of one of the equivalent sides of the triangle.

(Note, when applying the formula divide the measurement of the base in half--so the triangle will have sides of and .

Thus, the solution is:

$c=\sqrt{73}$

In this problem, the isosceles triangle will have two sides with the length of $\sqrt{73}$ and one side length of . Therefore, the perimeter is: $2\sqrt{73}+6$

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Question

A scalene triangle has sides with lengths of $\frac{3}{4}$ foot, $\frac{1}{4}$ foot, and $\frac{2}{6}$ foot. Find the perimeter of the the triangle (in inches).

Answer

To solve this problem, first convert each of the fractions from feet to inches.

$\frac{3}{4}=\frac{9}{12}=9$ inches

$\frac{1}{4}=\frac{3}{12}=3$ inches

$\frac{2}{6}=\frac{4}{12}=4$ inches

The perimeter of the triangle must equal

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Question

An acute triangle has side lengths of and . The hypotenuse of the triangle is not . Find the perimeter of this triangle.

Answer

Since the two side lengths provided in the question are not the hypotenuse, first use the Pythagorean Theorem to find the length of the third side.

$c=\sqrt{369}=\sqrt{9}\sqrt{41}=3\sqrt{41}$

The perimeter is equal to .

$12+15+3\sqrt{41}$

$3\sqrt{41}+27$

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Question

Find the perimeter of the triangle shown above.

Answer

Since the two side lengths provided in the question are not the hypotenuse, first use the Pythagorean Theorem to find the length of the third side.

$c=\sqrt{26.5}$

Perimeter is equal to .

$2.5+4.5+\sqrt{26.5}=7+ \sqrt{26.5}$

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Question

An obtuse Isosceles triangle has an area of square units and a base with length . Find the perimeter of this triangle.

Answer

To solve this solution, first work backwards using the formula: $Area=\frac{b\cdot h}{2}$

$32=\frac{8h}{2}$

Now that you've found the height of the triangle, use the Pythagorean Theorem to find the length of one of the equivalent sides of the triangle.

(Note, when applying the formula divide the measurement of the base in half--so the triangle will have sides of and .

Thus, the solution is:

$c=\sqrt{80}=4\sqrt{5}$

In this problem, the isosceles triangle will have two sides with the length of $4\sqrt{5}$ and one side length of . Therefore, the perimeter is: $8+8\sqrt{5}$ .

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Question

Find the perimeter of the acute isosceles triangle shown above. Note, the triangle has a base of and height of .

Answer

Use the Pythagorean Theorem to find the length of one of the equivalent sides of the triangle.

(Note, when applying the formula divide the measurement of the base in half--so the triangle will have sides of and .

Thus, the solution is:

$c=\sqrt{449}$

In this problem, the isosceles triangle will have two sides with the length of $\sqrt{449}$ and one side length of . Therefore, the perimeter is: $14+2\sqrt{449}$ .

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Question

For the triangle below, the perimeter is . Find the value of .

Answer

The dashes on the two sides of the triangle indicate that those two sides are congruent. Thus, the length of the missing side is also.

Now, use the information given about the perimeter to set up an equation to solve for . The sum of all the sides will give the perimeter.

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Question

Find the value of if the perimeter is .

Answer

The dashes on the two sides of the triangle indicate that those two sides are congruent. Thus, the length of the missing side is also.

Now, use the information given about the perimeter to set up an equation to solve for . The sum of all the sides will give the perimeter.

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Question

Find the value of if the perimeter is .

Answer

The dashes on the two sides of the triangle indicate that those two sides are congruent. Thus, the length of the missing side is also.

Now, use the information given about the perimeter to set up an equation to solve for . The sum of all the sides will give the perimeter.

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Question

Find the perimeter of the triangle below. Round to the nearest tenths place.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, where the long leg is the length of the short leg times $\sqrt3$ and the hypotenuse is two times the length of the short side. We can find out that the height of the triangle is since it is the short leg and the hypotenuse is .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $6, 6, \text{ and }6\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=6+6+6\sqrt3=22.4$

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Question

Find the perimeter of the triangle below. Round to the nearest tenths place.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, where the long leg is the short leg times $\sqrt 3$ and the hypotenuse is two times the short leg, we can find out that the height of the triangle is $\frac{3}{2}$ and the hypotenuse is .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $3, 3, \text{and }3\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=3+3+3\sqrt3=11.2$

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Question

Find the perimeter of the triangle below. Round to the nearest tenths place.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, where the long leg is the short leg times $\sqrt3$ and the hypotenuse is twice the length of the short leg, we can find out that the height of the triangle is and the hypotenuse is .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $8, 8, \text{ and }8\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=8+8+8\sqrt3=29.9$

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Question

In terms of , find the perimeter.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, we can find out that the height of the triangle is and the hypotenuse is .

The ratio of the sides in a triangle are: $x:x\sqrt3:2x$ .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $2x, 2x, \text{ and }2x\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=2x+2x+2x\sqrt3=4x+2x\sqrt3$

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Question

In terms of , find the perimeter.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, we can find out that the height of the triangle is and the hypotenuse is .

The ratio of the sides in a triangle are: $x:x\sqrt3:2x$ .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $42x, 42x, \text{ and }42x\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=42x+42x+42x\sqrt3=84x+42x\sqrt3$

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Question

Find the perimeter of the triangle below. Round to the nearest tenths place.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, we can find out that the height of the triangle is and the hypotenuse is .

The ratio of the sides in a triangle are: $x:x\sqrt3:2x$ .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $16, 16, \text{ and }16\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=16+16+16\sqrt3=59.7$

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Question

In terms of , find the perimeter of the triangle.

Answer

Draw in the height to create a right triangle.

Now, using the relationship between the lengths of sides in a triangle, we can find out that the height of the triangle is and the hypotenuse is .

The ratio of the sides in a triangle are: $x:x\sqrt3:2x$ .

The dashes on two sides of the triangle indicate that these two sides are congruent. The three side lengths of the triangle are $36x, 36x, \text{ and }36x\sqrt3$ .

Now, add up these side lengths to find the perimeter.

$\text{Perimeter}=36x+36x+36x\sqrt3=72x+36x\sqrt3$

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Question

A triangle is defined by the following points on a coordinate plane:

What is the perimeter of the triangle?

Answer

In order to find the perimeter of the triangle, we will first need to find the length of each side of the triangle by using the distance formula.

Recall the distance formula for a line:

$\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The first side of the triangle is the line segment made with $(1, 5)\text{ and }(2, 12)$ as its endpoints.

$\text{side 1}=\sqrt{(2-1)^2+(12-5)^2}=\sqrt{50}=5\sqrt2$

The second side of the triangle is the line segment that has $(2, 12)\text{ and }(-3, 4)$ as its endpoints.

$\text{side 2}=\sqrt{(2-(-3))^2+(12-4)^2}=\sqrt{89}$

The third side of the triangle is the line segment that has $(-3, 4)\text{ and }(1, 5)$ as its endpoints.

$\text{side 3}=\sqrt{(1-(-3))^2+(5-4)^2}=\sqrt{17}$

Now, add up these three sides with a calculator to find the perimeter of the triangle.

$\text{Perimeter}=5\sqrt{2} +\sqrt{89}+\sqrt{17}=20.63$

Make sure to round to places after the decimal.

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Question

A triangle is defined by the following points in a coordinate plane: .

What is the perimeter of the triangle?

Answer

In order to find the perimeter of the triangle, we will first need to find the length of each side of the triangle by using the distance formula.

Recall the distance formula for a line:

$\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The first side of the triangle is the line segment made with $(5, 6)\text{ and }(7, 2)$ as its endpoints.

$\text{side 1}=\sqrt{(7-5)^2+(2-6)^2}=\sqrt{20}=2\sqrt5$

The second side of the triangle is the line segment that has $(7, 2)\text{ and }(9, 11)$ as its endpoints.

$\text{side 2}=\sqrt{(9-7)^2+(11-2)^2}=\sqrt{85}$

The third side of the triangle is the line segment that has $(5, 6)\text{ and }(9, 11)$ as its endpoints.

$\text{side 3}=\sqrt{(9-5)^2+(11-6)^2}=\sqrt{41}$

Now, add up these three sides with a calculator to find the perimeter of the triangle.

$\text{Perimeter}=2\sqrt{5} +\sqrt{85}+\sqrt{41}=20.09$

Make sure to round to places after the decimal.

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Question

A triangle is defined by the following points on a coordinate plane: .

What is the perimeter of the triangle?

Answer

In order to find the perimeter of the triangle, we will first need to find the length of each side of the triangle by using the distance formula.

Recall the distance formula for a line:

$\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The first side of the triangle is the line segment made with $(-5, 1)\text{ and }(6, -2)$ as its endpoints.

$\text{side 1}=\sqrt{(6-(-5))^2+(-2-1)^2}=\sqrt{130}$

The second side of the triangle is the line segment that has $(6, -2)\text{ and }(7, 0)$ as its endpoints.

$\text{side 2}=\sqrt{(7-6)^2+(0-(-2))^2}=\sqrt{5}$

The third side of the triangle is the line segment that has $(-5, 1)\text{ and }(7, 0)$ as its endpoints.

$\text{side 3}=\sqrt{(7-(-5))^2+(0-1)^2}=\sqrt{145}$

Now, add up these three sides with a calculator to find the perimeter of the triangle.

$\text{Perimeter}=\sqrt{130} +\sqrt{5}+\sqrt{145}=25.68$

Make sure to round to places after the decimal.

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Question

A triangle is defined by the following points on a coordinate plane: .

What is the perimeter of the triangle?

Answer

In order to find the perimeter of the triangle, we will first need to find the length of each side of the triangle by using the distance formula.

Recall the distance formula for a line:

$\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The first side of the triangle is the line segment made with $(9, 8)\text{ and }(12, 0)$ as its endpoints.

$\text{side 1}=\sqrt{(12-9)^2+(0-8)^2}=\sqrt{73}$

The second side of the triangle is the line segment that has $(12, 0)\text{ and }(-1, -2)$ as its endpoints.

$\text{side 2}=\sqrt{(-1-12)^2+(-2-0)^2}=\sqrt{173}$

The third side of the triangle is the line segment that has $(9, 8)\text{ and }(-1, -2)$ as its endpoints.

$\text{side 3}=\sqrt{(-1-9))^2+(-2-8)^2}=\sqrt{200}=10\sqrt2$

Now, add up these three sides with a calculator to find the perimeter of the triangle.

$\text{Perimeter}=\sqrt{73} +\sqrt{173}+10\sqrt{2}=35.84$

Make sure to round to places after the decimal.

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