$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

Which of these could NOT be the perimeter of a right triangle with sides of length and ?

$11 + \sqrt{51}$

$11 + \sqrt{11}$

$11 + \sqrt{61}$

All of these work

None of these work.

Explanation

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.
Solve for a missing side using the Pythagorean theorem.
If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

$\textup{Perimeter}=a+b+c$

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

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

Simplify.

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

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

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

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

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

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

Screen shot 2016 07 07 at 12.58.14 pm

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Since it wasn't specified if the two lengths given were the legs, or a leg and the hypotenuse, there are 2 options for the third side length.

Option 1:

$\sqrt{61 } = c$

To find the perimeter, add the following:

$5+6+ \sqrt{61} = 11 + \sqrt{61}$

Option 2:

$\sqrt{11 } = b$

To find the perimeter, add the following:

$5 +6 + \sqrt{11} = 11 + \sqrt{11}$

The only option listed that isn't either of those is $11 + \sqrt{51}$ , so that is the answer.

A kite has a perimeter of inches. One pair of adjacent sides of the kite have a length of inches. What is the measurement for each of the other two sides of the kite?

$\frac{7}{2}in$

$\frac{6}{5}in$

Explanation

To find the missing side of this kite, work backwards using the formula:

, where and represent the length of one side from each of the two pairs of adjacent sides.

The solution is:

$\frac{26}{2}=(6+b)$

A sector in a circle with a radius of has an area of $12\pi$ . In degrees, what is the measurement of the central angle for this sector?

$30^\circ$

$45^\circ$

$60^\circ$

$15^\circ$

Explanation

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$

The central angle is degrees.

Find the area of a rhombus if the both diagonals have a length of $\sqrt{x+2}$ .

$\frac{x+2}{2}$

$\frac{\sqrt{2x+4}}{2}$

$\sqrt{2x+4}$

Explanation

Write the formula for the area of a rhombus.

$A=\frac{d_1 \cdot d_2}{2}$

Since both diagonals are equal, . Plug in the diagonals and reduce.

$A=\frac{(\sqrt x+2)(\sqrt x+2)}{2}=\frac{\sqrt{x+2}^2}{2}= \frac{x+2}{2}$

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent triangles.

Recall that the side lengths in a triangle are in a $1:\sqrt3:2$ ratio. Thus, the radius of the circle, which is also the base of the triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(3)}{3}=2\sqrt3$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(2\sqrt3)+\frac{\pi(2\sqrt3)}{2}=4\sqrt3+\pi\sqrt3=12.37$

Find the perimeter.

Explanation

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.
Solve for a missing side using the Pythagorean theorem.
If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

$\textup{Perimeter}=a+b+c$

Method 2:

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

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

Simplify.

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

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

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

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

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

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

Screen shot 2016 07 07 at 12.58.14 pm

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

The given triangle has of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of side , rewrite the equation.

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

Plug in the values of and .

$b=\sqrt{15^2-10^2}=\sqrt{125}$

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places.

$\text{Perimeter}=15+10+\sqrt{125}=36.18$

The height of a right triangle is 7 and the base is 12. What is the perimeter? Round to the nearest hundreth.

$\small 32.89$

$\small 13.89$

$\small 9.75$

$\small 28.75$

$\small 206.89$

Explanation

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.
Solve for a missing side using the Pythagorean theorem.
If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

Screen shot 2016 07 07 at 11.31.44 am

$\textup{Perimeter}=a+b+c$

Method 2:

Screen shot 2016 07 07 at 10.13.54 am

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

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

Simplify.

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

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

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

Simplify.

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

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

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

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

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

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

Screen shot 2016 07 07 at 12.58.14 pm

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem. In this problem, we've been given the two legs of the right triangle. In order to find the perimeter, we have to know the hypotenuse as well. In order to figure out that length, use the Pythagorean theorem:

$\small a^2 + b^2 = c^2$ where a and b are the lengths of the legs and c is the length of the hypotenuse.

$\small 7^2 + 12^ 2 = c^2$

$\small 49 + 144 = c^2$

$\small 193 = c^2$ take the square root of both sides

$\small \sqrt{193 } \approx 13. 89 = c$

To find the perimeter, just add $\small 7 + 12 + 13.89 = 32.89$

If the perimeter of a rectangle is , and the width of the rectangle is , what is the area of a rectangle?

Explanation

Recall how to find the perimeter of a rectangle:

$\text{Perimeter}=2(\text{length}+\text{width})$

Since we are given the width and the perimeter, we can solve for the length.

$\text{length}+\text{width}=\frac{\text{Perimeter}}{2}$

$\text{length}=\frac{\text{Perimeter}}{2}-\text{width}$

Substitute in the given values for the width and perimeter to find the length.

$\text{length}=\frac{36}{2}-17$

Simplify.

$\text{length}=18-17$

Solve.

$\text{length}=1$

Now, recall how to find the area of a rectangle.

$\text{Area}=\text{length}\times\text{width}$

Substitute in the values of the length and width to find the area.

$\text{Area}=17 \times 1$

Solve.

$\text{Area}=17$

A triangle is placed in a parallelogram so they share a base.

If the height of the triangle is half that of the parallelogram, find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

$\text{Area of Parallelogram}=\text{base}\times\text{height}$

$\text{Area of Parallelogram}=12\times8=96$

Next, recall how to find the area of a triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

Now, find the height of the triangle.

$\text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{2}$

$\text{Height of Triangle}=\frac{8}{2}=4$

Plug this value in to find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(12 \times 4)=24$

Subtract the two areas to find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Parallelogram}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=96-24=72$

Plane Geometry

Help Questions

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Explanation

How do you find the perimeter of a right triangle?

Method 1:

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Method 3:

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How do you find the perimeter of a right triangle?

Method 1:

Method 2:

Method 3:

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How do you find the perimeter of a right triangle?

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Method 3:

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