# 9.6.3. Solving Two Equations in Two Variables, One Is of the First Degree, and the Other Is of the Second Degree

In a right-angled triangle, the length of one of the two sides between which the right angle lies is cm less than that of the other, and the hypotenuse is cm long. Find the perimeter to the nearest centimetre.

• Acm
• Bcm
• Ccm
• Dcm

### Example

In a right-angled triangle, the length of one of the two sides between which the right angle lies is cm less than that of the other, and the hypotenuse is cm long. Find the perimeter to the nearest centimetre.

### Solution

Denote the two sides adjacent to the right angle as and , where is the longer of the two sides. Then it follows that

As the triangle is right-angled, it satisfies Pythagoras' theorem. Therefore,

This pair of equations can be solved by substitution. Substituting the first order equation into the second order equation gives

Dividing by 2 gives

This equation can be solved by factorisation.

As represents the length of one side of a triangle, it must have a positive value. Therefore, cm.

Now substitute this value of into the first order equation to find .

The perimeter of the triangle is therefore

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