Question

The hypotenuse of a right-angled triangle is 1 cm longer than twice the shorter side and is 2 cm longer than the other side.

Find the perimeter of the triangle.

Answer 1

40 cm

Explanation:

Pythagoras Theorem

`a^2+b^2=c^2`

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:

`\implies c = 2a + 1`

If the hypotenuse is 2 cm longer than the other side then:

`\implies c = b + 2`

Equate the two expressions for c and solve for b:

`\implies b+2=2a+1`

`\implies b=2a-1`

Substitute the expression for c involving a, and the expression for b involving a, into Pythagoras Theorem and solve for a:

`\implies a^2+(2a-1)^2=(2a+1)^2`

`\implies a^2+4a^2-4a+1=4a^2+4a+1`

`\implies a^2-4a=4a`

`\implies a^2-8a=0`

`\implies a(a-8)=0`

`\implies a=0, \quad a=8`

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:

`\begin{aligned} \implies \textsf{Perimeter}&=a+b+c\\&=a+(2a-1)+(2a+1)\\&=5a\end{aligned}`

Substitute the found value of a into the expression for the perimeter:

`\begin{aligned} \implies \textsf{Perimeter}&=5(8)\\&=40\sf \; cm\end{aligned}`

Therefore, the perimeter of the triangle is 40 cm.

Answer 2

• The perimeter is 40 cm

Explanation:

Let the sides be:

• Hypotenuse = a,
• Short leg = b,
• Long leg = c.

According to question are given:

• a = 2b + 1 ⇒ b = (a - 1)/2,
• a = c + 2 ⇒ c = a - 2.

Use Pythagorean to find the length of hypotenuse:

• a² = b² + c²

• a² = [(a - 1)/2]² + (a - 2)²
• a² = (a² - 2a + 1)/4 + a² - 4a + 4
• 0 = (a² - 2a + 1)/4 - 4a + 4
• a² - 2a + 1 - 16a + 16 = 0
• a² - 18a + 17 = 0
• a² - a - 17a + 17 = 0
• a(a - 1) - 17(a - 1) = 0
• (a - 1)(a - 17) = 0

• a - 1 = 0 and a - 17 = 0
• a = 1 and a = 17

The first option is discarded, as all sides should be positive. With a = 1 we get negative value for c.

So a = 17 cm.

Find the other sides:

• b = (17 - 1)/2 = 8 cm
• c = 17 - 2 = 15 cm

Find the perimeter:

• P = a + b + c = 17 + 8 + 15 = 40 cm