### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : How To Find The Area Of An Acute / Obtuse Triangle

Two sides of a scalene triangle measure 4 centimeters and 7 centimeters, and their corresponding angle measures 30 degrees. Find the area of the triangle.

**Possible Answers:**

**Correct answer:**

,

where and are the lengths of two sides and is the angle measure.

Plug in our given values:

### Example Question #1 : How To Find The Area Of An Acute / Obtuse Triangle

A scalene triangle has a base length and a corresponding altitude of . Give the area of the triangle in terms of .

**Possible Answers:**

**Correct answer:**

,

where is the base and is the altitude.

### Example Question #1 : How To Find The Area Of An Acute / Obtuse Triangle

What is the area of a triangle on the coordinate plane with its vertices on the points ?

**Possible Answers:**

**Correct answer:**

The base can be seen as the (vertical) line segment connecting and , which has length . The height is the pependicular distance from to the segment; since the segment is part of the -axis, this altitude is horizontal and has length equal to -coordinate .

The area of this triangle is therefore

.

### Example Question #1 : How To Find The Area Of An Acute / Obtuse Triangle

What is the area of a triangle on the coordinate plane with its vertices on the points ?

**Possible Answers:**

**Correct answer:**

The base can be seen as the (horizontal) line segment connecting and , the length of which is . The height is the pependicular distance from to the segment; since the segment is part of the -axis, this altitude is vertical and has a length equal to -coordinate .

The area of this triangle is therefore

.

### Example Question #57 : Triangles

Figure NOT drawn to scale.

is a right triangle with altitude . What percent of has been shaded gray?

Choose the closest answer.

**Possible Answers:**

**Correct answer:**

The altitude of a right triangle from the vertex of its right angle - which, here, is - divides the triangle into two triangles similar to each other as well as the large triangle.

The similarity ratio of to is the ratio of the lengths of their hypotenuses. The hypotenuse of the latter is 18; that of the former, from the Pythagorean Theorem, is

The similarity ratio is therefore . The ratio of their areas is the square of this, or

The area of is

of that of , so the choice closest to the correct percent is 25%.

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