Calculating the area of an acute / obtuse triangle - GMAT Quantitative Reasoning
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What is the area, to the nearest whole square inch, of a triangle with sides 12, 13, and 15 inches?
What is the area, to the nearest whole square inch, of a triangle with sides 12, 13, and 15 inches?
Use Heron's formula:
where 
, and
Use Heron's formula:
where
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Calculate the area of the triangle (not drawn to scale).
Calculate the area of the triangle (not drawn to scale).
In this problem, the base is 12 and the height is 6. Therefore:
In this problem, the base is 12 and the height is 6. Therefore:
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Note: Figure NOT drawn to scale.
What is the area of the above figure?
Note: Figure NOT drawn to scale.
What is the area of the above figure?
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 
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
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Give the area of a triangle on the coordinate plane with vertices 
.
Give the area of a triangle on the coordinate plane with vertices
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:
This can be illustrated by showing this triangle inscribed inside a rectangle whose vertices are
The area of the white triangle
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:
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Which of the following cannot be the measure of the vertex angle of an isosceles triangle?
Which of the following cannot be the measure of the vertex angle of an isosceles triangle?
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.
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
Since all of these measures fall in that range, the correct response is that all are allowed.
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What is the area of the triangle on the coordinate plane formed by the 
-axis and the lines of the equations 
and 
?
What is the area of the triangle on the coordinate plane formed by the
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
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
The lines intersect at
The intersection of
These lines intersect at
The intersection of
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
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What is the area of a triangle on the coordinate plane with its vertices on the points 
?
What is the area of a triangle on the coordinate plane with its vertices on the points
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
.
The vertical segment connecting
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Which of the following is the area of a triangle on the coordinate plane with its vertices on the points 
, where 
?
Which of the following is the area of a triangle on the coordinate plane with its vertices on the points
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
.
We can view the horizontal segment connecting
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