All ACT Math Resources
Example Questions
Example Question #1 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle
What is the length of side ? Round to the nearest hundredth.
Cannot be computed
Begin by filling in the missing angle for your triangle. Since a triangle has a total of degrees, you know that the missing angle is:
Draw out the figure:
This problem becomes incredibly easy! This is an isosceles triangle. Therefore, you know that is , because it is "across" from a degree angle—which matches the other degree angle!
Example Question #1 : 45/45/90 Right Isosceles Triangles
The area of an isosceles right triangle is . What is its height that is correlative and perpendicular to a side that is not the hypotenuse?
Recall that an isosceles right triangle is a triangle. That means that it looks like this:
This makes calculating the area very easy! Recall, the area of a triangle is defined as:
However, since for our triangle, we know:
Now, we know that . Therefore, we can write:
Solving for , we get:
This is the length of the height of the triangle for the side that is not the hypotenuse.
Example Question #2 : 45/45/90 Right Isosceles Triangles
What is the area of an isosceles right triangle that has an hypotenuse of length ?
Based on the information given, you know that your triangle looks as follows:
This is a triangle. Recall your standard triangle:
You can set up the following ratio between these two figures:
Now, the area of the triangle will merely be (since both the base and the height are ). For your data, this is:
Example Question #1 : Isosceles Triangles
Find the height of an isoceles right triangle whose hypotenuse is
To solve simply realize the hypotenuse of one of these triangles is of the form where s is side length. Thus, our answer is .
Example Question #1 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle
The area of an isosceles right triangle is . What is its height that is correlative and perpendicular to this triangle's hypotenuse?
Recall that an isosceles right triangle is a triangle. That means that it looks like this:
This makes calculating the area very easy! Recall, the area of a triangle is defined as:
However, since for our triangle, we know:
Now, we know that . Therefore, we can write:
Solving for , we get:
However, be careful! Notice what the question asks: "What is its height that is correlative and perpendicular to this triangle's hypotenuse?" First, let's find the hypotenuse of the triangle. Recall your standard triangle:
Since one of your sides is , your hypotenuse is .
Okay, what you are actually looking for is in the following figure:
Therefore, since you know the area, you can say:
Solving, you get: .
Example Question #1 : 45/45/90 Right Isosceles Triangles
What is the area of an isosceles right triangle with a hypotenuse of ?
Now, this is really your standard triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees, because the triangle is isosceles.
Based on the description of your triangle, you can draw the following figure:
This is derived from your reference triangle for the triangle:
For our triangle, we could call one of the legs . We know, then:
Thus, .
The area of your triangle is:
For your data, this is:
Example Question #2 : 45/45/90 Right Isosceles Triangles
What is the area of an isosceles right triangle with a hypotenuse of ?
Now, this is really your standard triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees because the triangle is isosceles.
Based on the description of your triangle, you can draw the following figure:
This is derived from your reference triangle for the triangle:
For our triangle, we could call one of the legs . We know, then:
Thus, .
The area of your triangle is:
For your data, this is:
Example Question #1 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle
What is the area of an isosceles right triangle with a hypotenuse of ?
Now, this is really your standard triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees because the triangle is isosceles.
Based on the description of your triangle, you can draw the following figure:
This is derived from your reference triangle for the triangle:
For our triangle, we could call one of the legs . We know, then:
Thus, .
The area of your triangle is:
For your data, this is:
Example Question #4 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle
is a right isosceles triangle with hypotenuse . What is the area of ?
Right isosceles triangles (also called "45-45-90 right triangles") are special shapes. In a plane, they are exactly half of a square, and their sides can therefore be expressed as a ratio equal to the sides of a square and the square's diagonal:
, where is the hypotenuse.
In this case, maps to , so to find the length of a side (so we can use the triangle area formula), just divide the hypotenuse by :
So, each side of the triangle is long. Now, just follow your formula for area of a triangle:
Thus, the triangle has an area of .
Example Question #1 : Isosceles Triangles
What is the perimeter of an isosceles right triangle with an hypotenuse of length ?
Your right triangle is a triangle. It thus looks like this:
Now, you know that you also have a reference triangle for triangles. This is:
This means that you can set up a ratio to find . It would be:
Your triangle thus could be drawn like this:
Now, notice that you can rationalize the denominator of :
Thus, the perimeter of your figure is: