All ACT Math Resources
Example Questions
Example Question #2 : How To Find The Perimeter Of A 45/45/90 Right Isosceles Triangle
What is the perimeter of an isosceles right triangle with an area of ?
Recall that an isosceles right triangle is also a triangle. Your reference figure for such a shape is:
or
Now, you know that the area of a triangle is:
For this triangle, though, the base and height are the same. So it is:
Now, we have to be careful, given that our area contains . Let's use , for "side length":
Thus, . Now based on the reference figure above, you can easily see that your triangle is:
Therefore, your perimeter is:
Example Question #202 : Plane Geometry
A tree is feet tall and is planted in the center of a circular bed with a radius of feet. If you want to stabalize the tree with ropes going from its midpoint to the border of the bed, how long will each rope measure?
This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem giving us . If then .
Example Question #141 : Triangles
An isosceles right triangle has a hypotenuse of length . What is the perimeter of this triangle, in terms of ?
The ratio of sides to hypotenuse of an isosceles right triangle is always . With this in mind, setting as our hypotenuse means we must have leg lengths equal to:
Since the perimeter has two of these legs, we just need to multiply this by and add the result to our hypothesis:
So, our perimeter in terms of is:
Example Question #1 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle
A 44/45/90 triangle has a hypotenuse of . Find the length of one of its legs.
Cannot be determined
It's helpful to remember upon coming across a 45/45/90 triangle that it's a special right triangle. This means that its sides can easily be calculated by using a derived side ratio:
Here, represents the length of one of the legs of the 45/45/90 triangle, and represents the length of the triangle's hypotenuse. Two sides are denoted as congruent lengths () because this special triangle is actually an isosceles triangle. This goes back to the fact that two of its angles are congruent.
Therefore, using the side rules mentioned above, if , this problem can be resolved by solving for the value of :
Therefore, the length of one of the legs is 1.
Example Question #2 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle
In a 45-45-90 triangle, if the hypothenuse is long, what is a possible side length?
If the hypotenuse of a 45-45-90 triangle is provided, its side length can only be one length, since the sides of all 45-45-90 triangles exist in a defined ratio of , where represents the length of one of the triangle's legs and represents the length of the triangle's hypotenuse. Using this method, you can set up a proportion and solve for the length of one of the triangle's sides:
Cross-multiply and solve for .
Rationalize the denominator.
You can also solve this problem using the Pythagorean Theorem.
In a 45-45-90 triangle, the side legs will be equal, so . Substitute for and rewrite the formula.
Substitute the provided length of the hypothenuse and solve for .
While the answer looks a little different from the result of our first method of solving this problem, the two represent the same value, just written in different ways.
Example Question #1 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle
In a triangle, if the length of the hypotenuse is , what is the perimeter?
1. Remember that this is a special right triangle where the ratio of the sides is:
In this case that makes it:
2. Find the perimeter by adding the side lengths together:
Example Question #4 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle
The height of a triangle is . What is the length of the hypotenuse?
Remember that this is a special right triangle where the ratio of the sides is:
In this case that makes it:
Where is the length of the hypotenuse.
Example Question #1 : Triangles
Square has a side length of . What is the length of its diagonal?
Cannot be determined from the information provided
The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a triangle.
From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a triangle.
1) Using the Pythagorean Theorem
Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula: , where and are the lengths of the legs of the triangle, and is the length of the triangle's hypotenuse.
In this case, . We can substitute these values into the equation and then solve for , the hypotenuse of the triangle and the diagonal of the square:
The length of the diagonal is .
2) Using Properties of Triangles
The second approach relies on recognizing a triangle. Although one could solve this rather easily with Pythagorean Theorem, the following method could be faster.
triangles have side length ratios of , where represents the side lengths of the triangle's legs and represents the length of the hypotenuse.
In this case, because it is the side length of our square and the triangles formed by the square's diagonal.
Therefore, using the triangle ratios, we have for the hypotenuse of our triangle, which is also the diagonal of our square.
Example Question #1 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem
What is the length of the hypotenuse of an isosceles right triangle with an area of ?
Recall that an isosceles right triangle is also a triangle. It has sides that appear as follows:
Therefore, the area of the triangle is:
, since the base and the height are the same.
For our data, this means:
Solving for , you get:
So, your triangle looks like this:
Now, you can solve this with a ratio and easily find that it is . You also can use the Pythagorean Theorem. To do the latter, it is:
Now, just do your math carefully:
That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:
Example Question #4 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem
When the sun shines on a pole, it leaves a shadow on the ground that is also . What is the distance from the top of the pole to the end of its shadow?
The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse. . In this case, . Therefore, we do . So .