### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #51 : Triangles

Note: Figure NOT drawn to scale.

What is the measure of angle

**Possible Answers:**

**Correct answer:**

The two angles at bottom are marked as congruent. One forms a linear pair with a angle, so it is supplementary to that angle, making its measure . Therefore, each marked angle measures .

The sum of the measures of the interior angles of a triangle is , so:

### Example Question #12 : Acute / Obtuse Triangles

Which of the following is true about a triangle with two angles that measure and ?

**Possible Answers:**

This triangle is scalene and right.

This triangle is scalene and obtuse.

This triangle is isosceles and right.

This triangle cannot exist.

This triangle is isosceles and obtuse.

**Correct answer:**

This triangle cannot exist.

A triangle must have at least two acute angles; however, a triangle with angles that measure and could have at most one acute angle, an impossible situation. Therefore, this triangle is nonexistent.

### 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 #2 : 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 #5 : How To Find The Area Of An Acute / Obtuse Triangle

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%.

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

Which of the following could be the lengths of the three sides of a scalene triangle?

**Possible Answers:**

**Correct answer:**

A scalene triangle, by definition, has three sides of different lengths. We can identify the scalene triangle by converting measures to the same unit. We will convert to inches in this case.

5 feet = inches. This triangle can be eliminated, since two sides have length 60 inches.

7 feet = inches. This triangle can be eliminated, since two sides have length 84 inches.

These three measures are equal. This is an equilateral triangle, and it can be eliminated.

feet = inches. This triangle can be eliminated, since two sides have length 66 inches.

5 feet = 60 inches.

6 feet = inches.

The sides of this triangle measure 60, 72, and 84 inches, so the triangle is scalene.

### Example Question #1 : Equilateral Triangles

For an equilateral triangle, Side A measures and Side B measures . What is the length of Side A?

**Possible Answers:**

8

3

10

4

**Correct answer:**

8

First you need to recognize that for an equilateral triangle, all 3 sides have equal lengths.

This means you can set the two values for Side A and Side B equal to one another, since they measure the same length, to solve for .

You now know that , but this is not your answer. The question asked for the length of Side A, so you need to plug 3 into that equation.

So the length of Side A (and Side B for that matter) is 8.

### Example Question #2 : How To Find The Length Of The Side Of An Equilateral Triangle

One angle of an equilateral triangle is 60 degrees. One side of that triangle is 12 centimeters. What are the measures of the two other angles and two other sides?

**Possible Answers:**

55 degrees, 65 degrees, 12 centimeters, 13 centimeters

45 degrees, 75 degrees, 12 centimeters, 13 centimeters

90 degrees, 30 degrees, 12 centimeters, 12 centimeters

60 degrees, 60 degrees, 12 centimeters, 12 centimeters

50 degrees, 70 degrees, 5 centimeters, 12 centimeters

**Correct answer:**

60 degrees, 60 degrees, 12 centimeters, 12 centimeters

An equilateral triangle is one in which all three sides are congruent. It also has the property that all three interior angles are equal. In other words, all three angles of an equilateral triangle are always 60°. Since all sides are congruent, the other two sides both measure 12 centimeters.

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