# ACT Math : Geometry

## Example Questions

### Example Question #71 : Plane Geometry

A right triangle has integer sides with a ratio of , measured in . What is the smallest possible area of this triangle?

Explanation:

The easiest way to find the smallest possible integer sides is to simply factor the ratio we are given. In this case,  is already prime (since  is a prime number), so the smallest possible sides which hold to this triangle are  and . You may also recognize this number as a common Pythagorean triple.

The area of a triangle is expressed as , where  is the length and  is the height. Since our triangle is right, we know that two lines intersect at a  angle and thus serve well as our length and height. We also know that the longest side is always the hypotenuse, so the other two sides must be  and .

Applying our formula, we get:

Thus, the smallest possible area for our triangle is .

### Example Question #72 : Plane Geometry

Right triangle  has hypotenuse  cm and . Find the area of the triangle, in cm2, by using .

Round angles to four significant figures. Round side lengths to the nearest integer.

Explanation:

To find the area of a right triangle, find the lengths of the two perpendicular legs (since this gives us our length and height for the area formula).

In this case, we know that one angle is , and SOHCAHTOA tells us that , so:

Substitute the angle measure and hypotenuse into the formula.

Isolate the variable.

Solve the left side (rounding to the nearest integer) using our Pythagorean formula:

--->  Substitute known values.

--->  Simplify.

Square root both sides.

So with our two legs solved for, we now only need to apply the area formula for triangles to get our answer:

So, the area of our triangle is .

### Example Question #73 : Plane Geometry

Find the area of a right triangle whose height is 4 and base is 5.

Explanation:

To solve, simply use the formula for the area of a triangle given height h and base B.

Substitute

and  into the area formula.

Thus,

### Example Question #14 : How To Find The Area Of A Right Triangle

A right triangle has a total perimeter of 12, and the length of its hypotenuse is 5. What is the area of this triangle?

12

6

15

10

3

6

Explanation:

The area of a triangle is denoted by the equation 1/2 b x h.

b stands for the length of the base, and h stands for the height.

Here we are told that the perimeter (total length of all three sides) is 12, and the hypotenuse (the side that is neither the height nor the base) is 5 units long.

So, 12-5 = 7 for the total perimeter of the base and height.

7 does not divide cleanly by two, but it does break down into 3 and 4,

and 1/2 (3x4) yields 6.

Another way to solve this would be if you recall your rules for right triangles, one of the very basic ones is the 3,4,5 triangle, which is exactly what we have here

### Example Question #1 : How To Find The Perimeter Of A Right Triangle

In the figure below, right triangle  has a hypotenuse of 6. If  and , find the perimeter of the triangle .

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.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. 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:

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:

## 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  angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the  angle and they are labeled  and . The side of the triangle that is opposite of the  angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

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.

Simplify.

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.

Simplify.

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.

Simplify.

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:

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:

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:

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.

First, we need to use the Pythagorean theorem to solve for .

Because we are dealing with a triangle, the only valid solution is  because we can't have negative values.

After you have found , plug it in to find the perimeter. Remember to simplify all square roots!

### Example Question #2 : How To Find The Perimeter Of A Right Triangle

Find the perimeter of the triangle below.

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.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. 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:

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:

## 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  angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the  angle and they are labeled  and . The side of the triangle that is opposite of the  angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

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.

Simplify.

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.

Simplify.

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.

Simplify.

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:

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:

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:

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. The perimeter of a triangle is simply the sum of its three sides. Our problem is that we only know two of the sides.  The key for us is the fact that we have a right triangle (as indicated by the little box in the one angle).  Knowing two sides of a right triangle and needing the third is a classic case for using the Pythagorean theorem.  In simple (sort of), the Pythagorean theorem says that sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse.

Every right triangle has three sides and a right angle.  The side across from the right angle (also the longest) is called the hypotenuse.  The other two sides are each called legs.  That means in our triangle, the side with length 17 is the hypotenuse, while the one with length 8 and the one we need to find are each legs.

What the Pythagorean theorem tells us is that if we square the lengths of our two legs and add those two numbers together, we get the same number as when we square the length of our hypotenuse.  Since we don't know the length of our second leg, we can identify it with the variable .

This allows us to create the following algebraic equation:

which simplified becomes

To solve this equation, we first need to get the variable by itself, which can be done by subtracting 64 from both sides, giving us

From here, we simply take the square root of both sides.

Technically,  would also be a square root of 225, but since a side of a triangle can only have a positive length, we'll stick with 15 as our answer.

But we aren't done yet.  We now know the length of our missing side, but we still need to add the three side lengths together to find the perimeter.

### Example Question #3 : How To Find The Perimeter Of A Right Triangle

Given that two sides of a right triangle are  and  and the hypotenuse is unknown, find the perimeter of the triangle.

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.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. 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:

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:

## 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  angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the  angle and they are labeled  and . The side of the triangle that is opposite of the  angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

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.

Simplify.

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.

Simplify.

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.

Simplify.

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:

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:

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:

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 calculate the perimeter we need to find the length of the hypotenuse using the Pythagorean theorem.

Rearrange.

Substitute in known values.

Now that we have found the missing side, we can substitute the values into the perimeter formula and solve.

### Example Question #74 : Plane Geometry

Given a right triangle with a leg length of  and a hypotenuse length of , what is the height of the triangle?

Explanation:

1. Use the pythagorean theorem with  and  :

2. Solve for :

(Notice that this right triangle is also a  triangle.)

### Example Question #75 : Plane Geometry

A  ladder is leaning against a wall. If the bottom of the ladder touches the ground  from the base of the wall, approximately (to the nearest whole number) how far is the top of the ladder from the base of the wall?

Explanation:

To answer this, we first need to understand that we are trying to find the leg of a right triangle. The ladder, when leaned against the wall, forms a right triangle where the ladder is the hypotenuse, the wall is one leg, and the ground between the ladder and the wall is the other leg.

To find how high up the ladder touches the wall, we use the Pythagorean Theorem, which is:

is the length of the hypotenuse (the ladder itself),  is the distance of the ladder base from the wall, and  is how high the ladder touches on the wall.

We then rearrange to solve for one of the legs by subtracting  from both sides.

We can now plug in our values of 10 for the hypotenuse and one of our legs (in this case, )

We can then take the square root of this equation to get an answer for

The question asked us to approximate, so we must round to the nearest whole number. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

### Example Question #71 : Plane Geometry

Right triangle  has sides that are integers, and an area of . Which of the following could not be the height of ?