The hypotenuse of a right triangle
is six feet.
Looking at the right triangle below,
this side is the hypotenuse
because it is the longest side,
and it's also the side
opposite the right angle.
Let's label the length six feet.
Next the height is three
more than two times
the length of the base.
Determine the length of
the base and the height.
Round to the hundredths.
So the height or the length of this side
is three more than two times
the length of the base,
which is this side.
Let's label the length of the base x feet.
And because the height
is three more than two times
the length of the base,
we can use the expression two x plus three
for the length of this
side here, the height.
Again whatever x is,
the expression two x plus three
is three more than two times x.
And now to determine the length
of the base and the height,
we can use the Pythagorean Theorem
to set up and solve an equation.
Where the Pythagorean Theorem
is a squared plus b
squared equals c squared,
where a and b are the
lengths of the two legs,
which are the two sides
that form the right angle,
and c is the hypotenuse.
Let's let this side be
a and this side be b
and applying in the
Pythagorean Theorem we have
the square of x
plus the square of
the quantity two x plus three
equals the square of six.
Now let's simplify both
sides of the equation.
Square of x is equal to x squared.
To square the quantity two x plus three,
you will have two factors
of two x plus three,
and six squared equals 36.
To multiply the two binomials,
you will have four products,
one, two, three, and four.
We have x squared
plus two x times two x is four x squared
plus two x times three, that's plus six x,
plus three times two x, that's plus six x,
and then plus three times
three which is nine,
equals 36.
And now let's combine
like terms on the left.
We have two x squared terms,
and we have two x terms.
x squared plus four x squared,
or one x squared plus four
x squared is five x squared,
six x plus six x is
12x, giving us plus 12x,
and we have plus nine equals 36.
We have a quadratic equation.
Let's set the right side of
the equation equal to zero
by subtracting 36 on both sides.
Simplifying we have five x squared
plus 12x,
nine minus 36 equals negative 27,
giving us minus 27,
equals on the right
the difference is zero.
This does not factor,
and therefore we will need
to use the Quadratic Formula,
where a, the coefficient of
x squared is equal to five,
b, the coefficient of x is equal to 12,
and c, the constant term
is equal to negative 27.
Applying the Quadratic Formula,
we have x equals,
in the numerator we have
negative b which is negative 12
plus or minus the square root
of b squared which is 12 squared
minus four times a times c
which is minus four times
five times a negative 27,
all divided by two times
a which is two times five.
Let's continue simplifying
on the next slide.
In the numerator we have negative 12
plus or minus the square root
of 12 squared is 144,
then we have minus four
times five times negative 27,
which is minus negative 540,
which is equivalent to plus 540.
All this is divided by
two times five which is 10.
Giving us x equals negative 12
plus or minus the square root
of 144 plus 540
equals 684,
all divided by 10.
Because of the plus or minus
we do have two solutions,
where one solution is x equals
the quantity negative 12 plus
the square root of 684
divided by 10,
and the other solution is x equals
negative 12 minus
the square root of 684
divided by 10.
But keep in mind,
x is a length, and therefore,
for the application problem,
we can only use the positive value for x.
We are asked to round to
the hundredths place value,
so now let's go to the calculator.
This value of x is going to be negative,
so we will exclude it,
but let's still determine
the approximate value
to the hundredths place value.
We do need parentheses
around the numerator,
so we start with an open parenthesis,
and then negative 12
minus the square root
of 684,
right arrow to get out from
underneath the square root,
close parenthesis, and
then divide it by 10.
Enter.
Rounding to the hundredths place value,
notice how there's a five in
the thousandths place value,
which means you round up.
X is approximately negative 3.82.
But again we do exclude this value because
we know x is a length,
and a length must be positive.
And now let's determine this value of x
to the hundredths place value.
If we press second Enter,
we get the previous entry
which we can now edit.
We can press the left arrow
and change the minus to plus.
And then press Enter.
To the hundredths place value,
x is approximately 1.42.
So going back to the first slide,
we now know the base has a length
of approximately 1.42 feet.
Now in determining the height,
we don't want to use
this rounded value of x
in the expression two x plus three.
We want to use the exact value of x,
and then round after determining
the value of the expression
two x plus three.
Because of this,
let's go back to the calculator.
One way to do this
is to store this value here
in the variable x
and then evaluate the
expression two x plus three.
To do this, we press the
Store button which is here,
and then the variable x,
and then Enter.
And now that value is
stored for the value of x,
so now we can just enter the
expression two x plus three,
Enter,
and round to the hundredths place value.
Two x plus three is approximately 5.83.
Which means the height is
approximately 5.83 feet.
Again if we use the
approximate value of x as 1.42
to perform the calculation
for two x plus three,
there will be more of an error,
and it may be marked wrong.
I hope you found this helpful.
