A right angle triangle and a
rectangle are joined to make a compound shape. All measurements are in
centimetres. The shaded area of the rectangle is
48.75 centimetres squared. Find the perimeter of the compound
shape in centimetres, giving your answer to four significant figures. You must show all of your
The perimeter of the compound shape
is made up of the four orange lines and the three pink lines, as shown on the
diagram. Our first step is to calculate the
value of 𝑥. We’re told in the question that the
area of the rectangle is 48.75 centimetres squared. And we can calculate the area of
any rectangle by multiplying its length by its width. In this case, 𝑥 minus two
multiplied by 𝑥 minus three is equal to 48.75.
Our next step is to expand the
double brackets using the FOIL method. Multiplying the first terms, 𝑥
multiplied by 𝑥 gives us 𝑥 squared. Multiplying the outside terms gives
us negative three 𝑥. Multiplying the inside terms gives
us negative two 𝑥. And finally, multiplying the last
terms gives us positive six.
Negative two multiplied by negative
three is equal to positive six. Grouping the like terms here gives
us 𝑥 squared minus five 𝑥 plus six is equal to 48.75. Subtracting 48.75 from both sides
of this equation gives us 𝑥 squared minus five 𝑥 minus 42.75 is equal to zero.
We now have a quadratic equation
equal to zero that we can solve using the quadratic formula. This will give us two
solutions. 𝑥 is equal to negative 𝑏 plus or
minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎.
In our question, 𝑎 is equal to
one, the coefficient of 𝑥 squared; 𝑏 is equal to negative five, the coefficient of
𝑥; and 𝑐 is equal to negative 42.75. Substituting these into the formula
gives us 𝑥 is equal to negative negative five plus minus the square root of
negative five squared minus four multiplied by one multiplied by negative 42.75 all
divided by two multiplied by one.
This simplifies to five plus or
minus the square root of 196 divided by two. The square root of 196 is 14. Therefore, either 𝑥 is equal to
five plus 14 divided by two or 𝑥 is equal to five minus 14 divided by two. This gives us two possible values
of 𝑥: 19 over two or negative nine over two. If you prefer, you could think of
these as decimals: 9.5 and negative 4.5. As 𝑥 is the base and height of the
triangle, it cannot be negative. Therefore, our value for 𝑥 is 19
over two, or 9.5.
We mentioned at the beginning of
the question that the perimeter of the compound shape was equal to the sum of the
four orange lines and the three pink lines. It’s important to note that the
three pink lines are equal to the length of the hypotenuse of the triangle. This means that the perimeter of
the shape can be written as 𝑥 plus 𝑥 plus ℎ plus 𝑥 minus three plus 𝑥 minus
This can be simplified to four 𝑥
plus ℎ minus six. Well, we know the value of 𝑥 is
now 19 over two, or 9.5. Therefore, we are left with four
multiplied by 19 over two plus ℎ minus six. We can calculate the value of ℎ
using Pythagoras’s theorem. 𝑎 squared plus 𝑏 squared equals
𝑐 squared, where 𝑐 is the hypotenuse or longest side of a right angle
triangle. In our question, this means that 𝑥
squared plus 𝑥 squared is equal to ℎ squared.
Simplifying this gives us two 𝑥
squared is equal to ℎ squared. Substituting in our value of 𝑥, 19
over two, gives us two multiplied by 19 over two squared equals ℎ squared. ℎ squared is, therefore, equal to
361 over two, or 180.5. Square-rooting both sides of this
equation gives us a value for ℎ of 19 root two over two. This is equal to 13.435 to three
decimal places. Substituting in this value of ℎ
enables us to work out the overall perimeter. The perimeter of the compound shape
is equal to 45.43502884.
Our final step is to round our
answer. We were asked to give our answer to
four significant figures. As the five after the dotted line
is greater than or equal to five, we need to round up. Therefore, our final answer is
45.44 centimetres to four significant figures.