A bakery fits a triangular metal brace under a shop shelf. The brace is right-angled, with the two shorter sides measuring \(x\) cm and \((x+7)\) cm, and th...
Question 1 Report
A bakery fits a triangular metal brace under a shop shelf. The brace is right-angled, with the two shorter sides measuring \(x\) cm and \((x+7)\) cm, and the hypotenuse measuring \((x+8)\) cm, as shown.
- Use Pythagoras' theorem to show that \(x^{2}-2x-15=0\). (2)
- Solve the equation to find the length of the hypotenuse, rejecting any solution that is not valid. (2)
Answer Details
In a right-angled triangle, Pythagoras' theorem states that the square of the hypotenuse equals the sum of the squares of the two shorter sides; writing this out algebraically for the given expressions, then expanding and simplifying, produces the required quadratic equation.
(a) The hypotenuse is \((x+8)\), so Pythagoras' theorem gives \(x^{2}+(x+7)^{2}=(x+8)^{2}\). Expanding both brackets: \(x^{2}+x^{2}+14x+49=x^{2}+16x+64\), which simplifies to \(2x^{2}+14x+49=x^{2}+16x+64\). Subtracting \(x^{2}+16x+64\) from both sides: \(x^{2}-2x-15=0\), as required. [2 marks]
(b) Factorising: \((x-5)(x+3)=0\), so \(x=5\) or \(x=-3\). Since \(x\) is a length, it cannot be negative, so \(x=-3\) is rejected; taking \(x=5\), the hypotenuse is \(x+8=13\) cm. [2 marks]
Rejecting the negative root is essential whenever a quadratic models a physical length: \(x=-3\) is a perfectly valid solution to the equation itself, but not to the original problem, since a side of a triangle cannot have negative length.
Download The App On Google Playstore
Everything you need to excel in your exams
