The difference between the two perpendicular sides of a right-angled triangle is 17 cm and its area is 84 cm². What is the perimeter (in cm) of the triangle?

The correct answer is 56 cm.

Let the two perpendicular sides (legs) of the right-angled triangle be a and b, where a > b.

We are given two conditions:

Condition 1: The difference between the two perpendicular sides is 17 cm.

a-b=17

Condition 2: The area of the triangle is 84 cm².

\frac{1}{2}\times a\times b=84

a\times b=168

Step 1: Express a in terms of b using Condition 1.

a=b+17

Step 2: Substitute into the area equation.

(b+17)\times b=168

b^2+17b-168=0

Step 3: Solve the quadratic equation by factoring.

We need two numbers that multiply to -168 and add to +17. Those numbers are +24 and -7.

(b+24)(b-7)=0

Since b must be positive, we take: b=7\text{cm}

Step 4: Find the value of a.

a=b+17=7+17=24\text{cm}

Verification of area: \frac{1}{2}\times 24\times 7=84\text{cm}{}^2 ✓

Step 5: Find the hypotenuse using the Pythagorean theorem.

h=\sqrt{a^2+b^2}=\sqrt{{24}^2+7^2}=\sqrt{576+49}=\sqrt{625}=25\text{cm}

Step 6: Calculate the perimeter.

\text{Perimeter}=a+b+h=24+7+25=56\text{cm}

Therefore, the perimeter of the triangle is 56 cm.