# SAT II Math I : Analyzing Figures

## Example Questions

### Example Question #1 : Analyzing Figures

The sides of a triangle have lengths 6 yards, 18 feet, and 216 inches. Which of the following is true about this triangle?

The triangle is acute and equilateral.

This triangle is right and isosceles, but not equilateral.

This triangle is acute and scalene.

This triangle is right and scalene.

This triangle is acute and isosceles, but not equilateral.

The triangle is acute and equilateral.

Explanation:

One yard is equal to 3 feet; it is also equal to 36 inches. Therefore:

18 feet is equal to  yards,

and

216 feet is equal to  yards.

The three sides are congruent, making the triangle equilateral - and all equilateral triangles are acute.

### Example Question #2 : Analyzing Figures

Figures not drawn to scale

# The triangles above are similar. Given the measurements above, what is the length of side c?

inches

inches

inches

inches

inches

inches

Explanation:

You can find the length of c by first finding the length of the hypotenuse of the larger similar triangle and then setting up a ratio to find the hypotenuse of the smaller similar triangle.

You also could have found 10 by recognizing this triangle is a form of a 3-4-5 triangle.

### The hypotenuse of the bigger triangle is 10 inches.

Now that we know the length of the hypotenuse for the larger triangle, we can set up a ratio equation to find the hypotenuse of the smaller triangle.

cross multiply

### Example Question #3 : Analyzing Figures

If line 2 and line 3 eventually intersect when extended to the left which of the following could be true?

Cannot be determined

I and III

I, II, and III

I only

I and II

I only

Explanation:

Read the question carefully and notice that the image is deceptive: these lines are not parallel. So we cannot apply any of our rules about parallel lines. So we cannot infer II or III, those are only true if the lines are parallel. If we sketch line 2 and line 3 meeting we will form a triangle and it is possible to make a = e. One such solution is to make a and e 60 degrees.

### Example Question #4 : Analyzing Figures

What is the maximum number of distinct regions that can be created with 4 intersecting circles on a plane?

Explanation:

Try sketching it out.

Start with one circle and then keep adding circles like a venn diagram and start counting. A region is any portion of the figure that can be defined and has a boundary with another portion. Don't forget that the exterior (labeled 14) is a region that does not have exterior boundaries.

### Example Question #1 : Analyzing Figures

Note: Figure may not be drawn to scale

In rectangle  has length and width  and  respectively. Point  lies on line segment  and point  lies on line segment .  Triangle  has area , in terms of  and  what is the possible range of values for ?

cannot be determined

Explanation:

Notice that the figure may not be to scale, and points  and  could lie anywhere on line segments  and  respectively.

Next, recall the formula for the area of a triangle:

To find the minimum area we need the smallest possible values for  and .

To make  smaller we can shift points  and  all the way to point . This will make triangle  have a height of :

is the minimum possible value for the area.

To find the maximum value we need the largest possible values for  and . If we shift point  all the way to point  then the base of the triangle is  and the height is , which we can plug into the formula for the area of a triangle:

which is the maximum possible area of triangle