How to find the area of an acute / obtuse triangle - ISEE Upper Level Mathematics Achievement
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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.
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.
,
where 
and 
are the lengths of two sides and 
is the angle measure.
Plug in our given values:
where
Plug in our given values:
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A scalene triangle has a base length 
and a corresponding altitude of 
. Give the area of the triangle in terms of 
.
A scalene triangle has a base length
,
where 
is the base and 
is the altitude.
where
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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 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
.
The base can be seen as the (horizontal) line segment connecting
The area of this triangle is therefore
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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 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
.
The base can be seen as the (vertical) line segment connecting
The area of this triangle is therefore
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Figure NOT drawn to scale.
is a right triangle with altitude 
. What percent of has been shaded gray?
Choose the closest answer.
Figure NOT drawn to scale.
Choose the closest 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%.
The altitude of a right triangle from the vertex of its right angle - which, here, is
The similarity ratio of
The similarity ratio is therefore
The area of
of that of
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