A square has perimeter 1.

True or false: The area of the square is $\frac{1}{2}$ .

False

True

Explanation

All four sides of a square have the same length, so the common sidelength is one fourth of the perimeter. The perimeter of the given square is 1, so the length of each side is $\frac{1}{4}$ .

The area of a square is equal to the square of the length of a side, so the area of this square is

$A =\left ( \frac{1}{4} \right )^{2} = \frac{1^{2} }{4^{2} } = \frac{1}{16}$ .

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

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.

In order to find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

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

Plug in the values of the lengths of the legs of the given triangle.

$c=\sqrt{3^2+20^2}=\sqrt{409}$

Now, recall how to find the perimeter of a triangle:

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

Plug in all the values of the sides of the triangle to find the perimeter.

$\text{Perimeter}=3+20+\sqrt{409}=43.22$

Make sure to round to places after the decimal.

An isosceles triangle is placed in a circle as shown by the figure below.

If diameter of the circle is , find the area of the shaded region.

Explanation

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

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

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

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

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 1^2= \pi$

To find the area of the shaded region, subtract the two areas.

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

$\text{Area of Shaded Region}=\pi-1=2.14$

Make sure to round to places after the decimal.

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.

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

Rhombus_1

The above figure shows a rhombus . Give its area.

$40 \sqrt{11}$

$40\sqrt{61}$

Explanation

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

Rhombus_1

By the Pythagorean Theorem,

$n = \sqrt{12^{2}- 10^{2}} =\sqrt{144- 100} = \sqrt{44}= \sqrt{4}\cdot \sqrt{11} = 2 \sqrt{11}$

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and $2 \sqrt{11}$ , so the area of the rhombus is

$A = 4 \cdot \frac{1}{2}\cdot 10 \cdot 2\sqrt{11} =40 \sqrt{11}$ .

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into congruent triangles.

We can then use the height to find the length of the side of the triangle.

Recall that a triangle has sides that are in ratios of $1:\sqrt3:2$ . The smallest side in the given figure is the base, the second longest side is the height, and the longest side is the side of the triangle itself.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

$\text{side}=\frac{2\sqrt3(\text{2})}{3}=\frac{4\sqrt3}{3}$

Now, since the perimeter of the shape consists of of these sides, we can use the following equation to find the perimeter.

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{4\sqrt3}{3})=\frac{20\sqrt3}{3}=11.55$

If the diameter of the circle below is , what is the area of the shaded region?

Explanation

From the given figure, you should notice that the base of the triangle is the same as the diameter of the circle.

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

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi\times\text{radius}^2$

Now recall the relationship between the radius and the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Plug in the value of the diameter to find the value of the radius.

$\text{Radius}=\frac{8}{2}=4$

Now, plug in the value of the radius in to find the area of the circle.

$\text{Area of Circle}=\pi\times 4^2=16\pi$

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

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

The height is already given by the question, and remember that the base is the same as the diameter of the circle.

Plug in these values to find the area of the triangle.

$\text{Area of Triangle}=\frac{4 \times 8}{2}=16$

We are now ready to find the area of the shaded region.

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

$\text{Area of Shaded Region}=16\pi-16=34.27$

Remember to round to decimal places.

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we must first find the areas of the triangle and the rectangle.

Recall how to find the area of a rectangle:

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

Substitute in the given length and width to find the area.

$\text{Area of Rectangle}=8 \times 12$

$\text{Area of Rectangle}= 96$

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

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

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

$\text{height}=\frac{\text{length}}{2}$

$\text{height}=\frac{8}{2}$

$\text{height}=4$

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

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

$\text{Area of Triangle}=24$

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

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

Solve.

$\text{Area of Shaded Region}=72$

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 the hypotenuse, , rewrite the equation.

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

Plug in the values of and .

$c=\sqrt{5^2+10^2}=\sqrt{125}$

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

$\text{Perimeter}=5+10+\sqrt{125}=26.18$

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 the hypotenuse, , rewrite the equation.

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

Plug in the values of and .

$c=\sqrt{6^2+14^2}=\sqrt{232}$

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

$\text{Perimeter}=6+14+\sqrt{232}=35.23$

Basic Geometry

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