visionId":"bce3538e-d105-447f-987e-fe74c1e7e12f","questionNumber":null,"questionType":"LA","specification":"![A triangular decorative tile has two sides labeled x cm and (x + 6) cm, with an included angle of 45 degrees.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/babfc30d-d017-4fd2-8901-712131259a60/image-1-2c8f53e64354ee7de759b2b5.svg)","difficulty":4,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":"calc","isCalculation":true,"parts":[{"id":"fc5b46dd-3e6e-4f90-910d-121b921f0fed","label":"a","content":"The area of the tile is $20 \\text{ cm}^2$. \nWork out the value of $x$. \nGive your answer to 3 significant figures.","markscheme":"- Sets up equation: $\\frac{1}{2}(x)(x+6)\\sin(45) = 20$ M1\n- Simplifies to $x^2 + 6x = \\frac{40}{\\sin(45)}$ or $x^2 + 6x \\approx 56.568$ M1\n- Forms quadratic $x^2 + 6x - 56.568... = 0$ M1\n- Applies quadratic formula $x = \\frac{-6 \\pm \\sqrt{6^2 - 4(1)(-56.568)}}{2}$ M1\n- Calculates $x = 5.10$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up the area equation using 1/2 * x * (x+6) * sin(45) = 20.","code":"M"},{"id":"p2","marks":1,"description":"Expansion of brackets and division by sin(45) constant.","code":"M"},{"id":"p3","marks":1,"description":"Correct identification of quadratic coefficients for formula use.","code":"M"},{"id":"p4","marks":1,"description":"Substitution into quadratic formula or use of calculator solver.","code":"M"},{"id":"p5","marks":1,"description":"Final positive root for x given to 3 significant figures.","code":"A"}],"finalAnswers":["5.10"],"generatedAt":"2026-05-06T14:01:35.298Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"quadratic_formula","label":"Form and solve a quadratic","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$A=\\dfrac{1}{2}ab\\sin C$"},{"latex":"$$20=\\dfrac{1}{2}x(x + 6)\\sin 45^\\circ$"}]},"order":0,"title":"Use the area formula","description":"For two sides and their included angle, use the triangle area formula. Substituting the given measurements and area earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$20=\\dfrac{1}{2}x(x+6)\\sin 45^\\circ$"},{"latex":"$$40=x(x+6)\\color{@sky}{\\htmlData{anchor=sine_from}{\\sin 45^\\circ}}$"},{"latex":"$$x(x+6)=\\dfrac{40}{\\color{@sky}{\\htmlData{anchor=sine_to}{\\sin 45^\\circ}}}$"},{"latex":"$$x(x+6)=56.568542\\ldots$"},{"latex":"$$x^2+6x=56.568542\\ldots$"}],"connections":[{"to":"sine_to","from":"sine_from","color":"sky","label":"divide by sin 45°","style":"solid"}]},"order":1,"title":"Remove the multiplier","description":"Multiply both sides by 2, then divide by sin 45°. Reaching the expanded equation earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+6x=56.568542\\ldots$"},{"latex":"$$x^2+6x-56.568542\\ldots=0$"}]},"order":2,"title":"Form the quadratic","description":"Subtract the right-hand value from both sides so that one side is zero. This standard quadratic form earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=c_from}{\\color{@purple}{c=-56.568542\\ldots}}$"},{"latex":"$$x=\\dfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}$"},{"latex":"$$x=\\dfrac{-6\\pm\\sqrt{6^2-4(1)\\htmlData{anchor=c_to}{\\color{@purple}{(-56.568542\\ldots)}}}}{2(1)}$"}],"connections":[{"to":"c_to","from":"c_from","color":"purple","label":"substitute"}]},"order":3,"title":"Apply the formula","description":"Here, a is 1, b is 6 and c is −56.568542…. Correct substitution into the quadratic formula earns the fourth M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=5.097440\\ldots$"},{"latex":"$$x=-11.097440\\ldots$"},{"text":"Reject the negative root because it cannot represent a length."},{"latex":"$$x=5.10\\text{ cm}\\quad\\text{(to 3 significant figures)}$"}]},"order":4,"title":"Choose the valid root","description":"A length cannot be negative, so reject the negative solution. Keep

Finding the Area of Any Triangle

Question 10

Finding the Area of Any Triangle Questions

Finding the Area of Any Triangle

Question 10

Finding the Area of Any Triangle Questions