# GMAT Math : Acute / Obtuse Triangles

## Example Questions

### Example Question #354 : Geometry

What is the area, to the nearest whole square inch, of a triangle with sides 12, 13, and 15 inches?

None of the other answers are correct.

Explanation:

Use Heron's formula:

where , and

### Example Question #355 : Geometry

Calculate the area of the triangle (not drawn to scale).

Explanation:

In this problem, the base is 12 and the height is 6. Therefore:

### Example Question #356 : Geometry

Note: Figure NOT drawn to scale.

What is the area of the above figure?

Explanation:

The figure is a composite of a rectangle and a triangle, as shown:

The rectangle has area

The triangle has area

The total area of the figure is

### Example Question #357 : Geometry

Which of the following cannot be the measure of the vertex angle of an isosceles triangle?

Each of the other choices can be the measure of the vertex angle of an isosceles triangle.

Each of the other choices can be the measure of the vertex angle of an isosceles triangle.

Explanation:

The only restriction on the measure of the vertex angle of an isosceles triangle is the restriction on any angle of a triangle - that it fall between  and , noninclusive. If  is any number in that range, each base angle, the two being congruent, will measure , which will fall in the acceptable range.

Since all of these measures fall in that range, the correct response is that all are allowed.

### Example Question #358 : Geometry

What is the area of the triangle on the coordinate plane formed by the -axis and the lines of the equations  and  ?

Explanation:

The easiest way to solve this is to graph the three lines and to observe the dimensions of the resulting triangle. It helps to know the coordinates of the three points of intersection, which we can do as follows:

The intersection of  and the -axis - that is, the line  can be found with some substitution:

The lines intersect at

The intersection of  and the  -axis can be found the same way:

These lines intersect at

The intersection of  and   can be found via the substitution method:

The lines intersect at

The triangle therefore has these three vertices. It is shown below.

As can be seen, it is a triangle with base 9 and height 12, so its area is

### Example Question #359 : Geometry

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

Explanation:

The vertical segment connecting  and  can be seen as the base of this triangle; this base has length . The height is the perpendicular (horizontal) distance from  to this segment, which is 6, the same as the -coordinate of this point. The area of the triangle is therefore

.

### Example Question #360 : Geometry

Which of the following is the area of a triangle on the coordinate plane with its vertices on the points  , where  ?

Explanation:

We can view the horizontal segment connecting , and  as the base; its length wiill be . The height will be the perpendicular (vertical) distance to this segment from the opposite point , which is , the -coordinate; therefore, the area of the triangle will be half the product of these two numbers, or

.

### Example Question #31 : Acute / Obtuse Triangles

Give the area of a triangle on the coordinate plane with vertices .

Explanation:

This can be illustrated by showing this triangle inscribed inside a rectangle whose vertices are :

The area of the white triangle  is the one whose area we calculate. To do this, we need the area of the square:

The area of the red triangle:

The area of the green triangle:

And the area of the beige triangle:

The area of the white triangle will be as follows:

### Example Question #1 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side ?

Explanation:

By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.

Case 1:  is the greatest of the three sidelengths.

Then

Case 2:  is not the greatest of the three sidelengths - that is, 11 is.

Then , or, equivalently, .

Therefore, .

### Example Question #2 : Calculating The Length Of The Side Of An Acute / Obtuse Triangle

The sides of a triangle are 4, 8, and an integer . How many possible values does  have?