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Example Questions
Example Question #10 : Calculating The Height Of An Acute / Obtuse Triangle
Given: with
and
.
Construct the altitude of from
to a point
on
. What is the length of
?
is shown below, along with altitude
; note that
has been extended to a ray
to facilitate the location of the point
.
Since an exterior angle of a triangle has as its measure the sum of those of its remote interior angles,
By definition of an altitude, is perpendicular to
, making
a right triangle and
a 30-60-90 triangle. By the 30-60-90 Triangle Theorem, shorter leg
of
has half the length of hypotenuse
—that is, half of 48, or 24; longer leg
has length
times this, or
, which is the correct choice.
Example Question #11 : Calculating The Height Of An Acute / Obtuse Triangle
Given: with
and
.
Construct the altitude of from
to a point
on
. What is the length of
?
is shown below, along with altitude
.
Since , and
, by definition, is perpendicular to
,
is a 30-60-90 triangle. By the 30-60-90 Triangle Theorem,
, as the shorter leg of
, has half the length of hypotenuse
; this is half of 30, or 15.
Example Question #12 : Calculating The Height Of An Acute / Obtuse Triangle
Given: with
, construct two altitudes of
: one from
to a point
on
, and another from
to a point
on
. Which of the following is true of the relationship of the lengths of
and
?
The length of is nine-sixteenths that of
.
The length of is two-thirds that of
.
The length of is four-ninths that of
.
The length of is twice that of
.
The length of is three-fourths that of
.
The length of is three-fourths that of
.
The area of a triangle is one half the product of the length of any base and its corresponding height; this is , but it is also
. Set these equal, and note the following:
That is, the length of is three fourths that of that of
.
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