### All ISEE Middle Level Math Resources

## Example Questions

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

A triangle has a base of and an area of . What is the height?

**Possible Answers:**

**Correct answer:**

The area of a triangle is found by multiplying the base by the height and dividing by two:

In this problem we are given the base, which is , and the area, which is . First we write an equation using as our variable.

To solve this equation, first multply both sides by , becuase multiplication is the opposite of division and therefore allows us to eliminate the .

The left-hand side simplifies to:

The right-hand side simplifies to:

So our equation is now:

Next we divide both sides by , because division is the opposite of multiplication, so it allows us to isolate the variable by eliminating .

So the height of the triangle is .

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

Note: Figure NOT drawn to scale.

The above triangle has area 36 square inches. If , then what is ?

**Possible Answers:**

**Correct answer:**

The area of a triangle is one half the product of its base and its height - in the above diagram, that means

.

Substitute , and solve for .

### Example Question #1 : Triangles

Please use the following shape for the question.

What is the area of this shape?

**Possible Answers:**

**Correct answer:**

From this shape we are able to see that we have a square and a triangle, so lets split it into the two shapes to solve the problem. We know we have a square based on the 90 degree angles placed in the four corners of our quadrilateral.

Since we know the first part of our shape is a square, to find the area of the square we just need to take the length and multiply it by the width. Squares have equilateral sides so we just take 5 times 5, which gives us 25 inches squared.

We now know the area of the square portion of our shape. Next we need to find the area of our right triangle. Since we know that the shape below the triangle is square, we are able to know the base of the triangle as being 5 inches, because that base is a part of the square's side.

To find the area of the triangle we must take the base, which in this case is 5 inches, and multipy it by the height, then divide by 2. The height is 3 inches, so 5 times 3 is 15. Then, 15 divided by 2 is 7.5.

We now know both the area of the square and the triangle portions of our shape. The square is 25 inches squared and the triangle is 7.5 inches squared. All that is remaining is to added the areas to find the total area. Doing this gives us 32.5 inches squared.

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

What is the area of the triangle?

**Possible Answers:**

**Correct answer:**

Area of a triangle can be determined using the equation:

### Example Question #1 : Area Of A Triangle

Bill paints a triangle on his wall that has a base parallel to the ground that runs from one end of the wall to the other. If the base of the wall is 8 feet, and the triangle covers 40 square feet of wall, what is the height of the triangle?

**Possible Answers:**

**Correct answer:**

In order to find the area of a triangle, we multiply the base by the height, and then divide by 2.

In this problem we are given the base and the area, which allows us to write an equation using as our variable.

Multiply both sides by two, which allows us to eliminate the two from the left side of our fraction.

The left-hand side simplifies to:

The right-hand side simplifies to:

Now our equation can be rewritten as:

Next we divide by 8 on both sides to isolate the variable:

Therefore, the height of the triangle is .

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

The hypotenuse of a right triangle is 25 inches; it has one leg 15 inches long. Give its area in square feet.

**Possible Answers:**

**Correct answer:**

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

The legs are 15 and 20 inches long. Divide both dimensions by 12 to convert from inches to feet:

feet

feet

Now find half their product:

square feet

### Example Question #32 : Triangles

Note: Figure NOT drawn to scale.

What percent of the above figure is green?

**Possible Answers:**

The correct answer is not given among the other choices.

**Correct answer:**

The area of the entire rectangle is the product of its length and width, or

.

The area of the right triangle is half the product of its legs, or

The area of the green region is therefore the difference of the two, or

.

The green region is therefore

of the rectangle.

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

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of the green region to that of the white region.

**Possible Answers:**

The correct answer is not given among the other choices.

**Correct answer:**

The area of the entire rectangle is the product of its length and width, or

.

The area of the right triangle is half the product of its legs, or

The area of the green region is therefore the difference of the two, or

.

The ratio of the area of the green region to that of the white region is

That is, 11 to 4.

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

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

**Possible Answers:**

None of these

**Correct answer:**

The area of a triangle is found by multiplying the base times the height, divided by 2.

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

The fraction cannot be simplified.

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

If the angles of a triangle are equal to , , and , what is the value of ?

**Possible Answers:**

**Correct answer:**

Given that there are 180 degrees in a triangle,

Thus, 30 is the correct answer.