Answer:
40 cm
Step-by-step explanation:
Pythagoras Theorem
[tex]a^2+b^2=c^2[/tex]
where:
- a and b are the legs of the right triangle.
- c is the hypotenuse of the right triangle.
Let a be the shorter leg.
If the hypotenuse is 1 cm longer than twice the shorter side then:
[tex]\implies c = 2a + 1[/tex]
If the hypotenuse is 2 cm longer than the other side then:
[tex]\implies c = b + 2[/tex]
Equate the two expressions for c and solve for b:
[tex]\implies b+2=2a+1[/tex]
[tex]\implies b=2a-1[/tex]
Substitute the expression for c involving a, and the expression for b involving a, into Pythagoras Theorem and solve for a:
[tex]\implies a^2+(2a-1)^2=(2a+1)^2[/tex]
[tex]\implies a^2+4a^2-4a+1=4a^2+4a+1[/tex]
[tex]\implies a^2-4a=4a[/tex]
[tex]\implies a^2-8a=0[/tex]
[tex]\implies a(a-8)=0[/tex]
[tex]\implies a=0, \quad a=8[/tex]
Since the length of a side cannot be zero, a = 8.
The perimeter of a two-dimensional shape is the distance around the outside. Therefore, the perimeter of the triangle is the sum of its sides:
[tex]\begin{aligned} \implies \textsf{Perimeter}&=a+b+c\\&=a+(2a-1)+(2a+1)\\&=5a\end{aligned}[/tex]
Substitute the found value of a into the expression for the perimeter:
[tex]\begin{aligned} \implies \textsf{Perimeter}&=5(8)\\&=40\sf \; cm\end{aligned}[/tex]
Therefore, the perimeter of the triangle is 40 cm.
