Video Transcript
A right triangle has side lengths
of two 𝑥 centimeters, 𝑥 plus five centimeters, and two 𝑥 plus four
centimeters. Find the value of 𝑥 and calculate
the perimeter and area of the triangle.
We will begin by sketching the
right triangle. We are told that the triangle has
side lengths two 𝑥, 𝑥 plus five, and two 𝑥 plus four, where each of these is
measured in centimeters. And convention dictates that the
third side is the hypotenuse or longest side. As we are dealing with a right
triangle, we can use the Pythagorean theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or
hypotenuse.
Substituting in the values from our
triangle, we have two 𝑥 squared plus 𝑥 plus five squared is equal to two 𝑥 plus
four squared. Two 𝑥 multiplied by two 𝑥 is
equal to four 𝑥 squared. For the other two terms, we will
need to distribute the parentheses, otherwise known as expanding the brackets. We begin by squaring 𝑥 plus five,
and we can use the FOIL method. Multiplying the first terms gives
us 𝑥 squared. Multiplying the outer terms, we
have five 𝑥. When multiplying the inner terms,
we also have five 𝑥. Finally, multiplying the last terms
gives us 25. Collecting like terms, this
simplifies to 𝑥 squared plus 10𝑥 plus 25. 𝑥 plus five all squared is
therefore equal to 𝑥 squared plus 10𝑥 plus 25.
We can repeat this process when
squaring two 𝑥 plus four. Two 𝑥 plus four multiplied by two
𝑥 plus four is equal to four 𝑥 squared plus eight 𝑥 plus eight 𝑥 plus 16. This simplifies to four 𝑥 squared
plus 16𝑥 plus 16. We now have the equation four 𝑥
squared plus 𝑥 squared plus 10𝑥 plus 25 is equal to four 𝑥 squared plus 16𝑥 plus
16. We can subtract four 𝑥 squared
from both sides of the equation. We can then subtract 16𝑥 and 16
from both sides of the equation, giving us 𝑥 squared minus six 𝑥 plus nine is
equal to zero.
We now have a quadratic equation
equal to zero, which we can solve by factoring. 𝑥 squared minus six 𝑥 plus nine
is equal to 𝑥 minus three multiplied by 𝑥 minus three. This is because the sum of negative
three and negative three is negative six and their product is nine. For the factored form to be equal
to zero, then one of the factors must equal zero. As both of our parentheses are 𝑥
minus three, we know that 𝑥 minus three must equal zero. Adding three to both sides of this
equation gives us a value of 𝑥 equal to three. We have now answered the first part
of the question. 𝑥 is equal to three.
We were also asked to calculate the
perimeter and area of the triangle. In order to do this, we will
substitute 𝑥 equals three back in to the three expressions for the lengths of our
triangle. The three lengths in centimeters
were two 𝑥, 𝑥 plus five, and two 𝑥 plus four. As 𝑥 is equal to three, we have
two multiplied by three. The first length of our triangle is
equal to six centimeters. Next, we have three plus five. The second side length of our
triangle is eight centimeters. Finally, we have two multiplied by
three plus four. Two multiplied by three is equal to
six and adding four to this gives us 10. Therefore, the length of the
hypotenuse is 10 centimeters.
We have a right triangle with side
lengths six centimeters, eight centimeters, and 10 centimeters. We know that to calculate the
perimeter of any shape, we simply add the lengths around the outside. Six plus eight plus 10 is equal to
24. As the perimeter is a measure of
length, this is equal to 24 centimeters. We know that the area of any
triangle is equal to its base multiplied by its height divided by two. In this question, we need to
multiply eight by six and then divide our answer by two or halve it. Eight multiplied by six is 48, and
dividing this by two gives us 24. As area is measured in square
units, the area of our triangle is 24 square centimeters.
If a right triangle has side
lengths of two 𝑥 centimeters, 𝑥 plus five centimeters, and two 𝑥 plus four
centimeters, then the value of 𝑥 is equal to three, the perimeter of the triangle
is 24 centimeters, and the area is 24 square centimeters.
