":"Verify your algebraic solving using a graphics calculator or equation solver."}]}]},{"id":"288d3302-122f-4ae6-bc2e-ee4faf94fd3c:perpendicular_height","label":"Using Perpendicular Height","steps":[{"type":"text","order":0,"title":"Define the side lengths","content":"The question tells us that the ratio of the length $BC$ to the length $AC$ is $3:1$.\n\nSince we are given that $AC = x$ cm, we can write the length of the base $BC$ as:\n$$BC = 3x \\text{ cm}$$","highlights":[{"text":"ratio of the length BC to the length AC is 3:1","location":"specification"},{"text":"Let the length AC be x cm.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin(30^\\circ) = \\frac{\\text{Opposite}}{\\text{Hypotenuse}} = \\frac{\\htmlData{anchor=height}{h}}{\\htmlData{anchor=hypotenuse}{x}}$"},{"latex":"$$\\htmlData{anchor=height-eq}{h} = \\htmlData{anchor=hypotenuse-sub}{x} \\times \\sin(30^\\circ)$"},{"latex":"$$\\sin(30^\\circ) = \\frac{1}{2}$"},{"latex":"$$h = \\frac{1}{2}x$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"height"},{"color":"amber","style":"text-color","anchor":"hypotenuse"}],"connections":[{"to":"height-eq","from":"height","color":"sky","label":"solve for height","style":"solid"},{"to":"hypotenuse-sub","from":"hypotenuse","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Find the perpendicular height","description":"To find the perpendicular height $h$ from vertex $A$ down to the base $BC$, we use trigonometry on the right-angled triangle formed at the left."},{"type":"text","order":2,"title":"Set up the area formula","content":"We know that the area of any triangle is given by the formula:\n$$\\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height}$$\n\nSubstituting our known terms:\n- $\\text{base} = 3x$\n- $\\text{height} = \\frac{1}{2}x$\n- $\\text{Area} = 75\\text{ cm}^2$\n\nWe get the equation:\n$$\\frac{1}{2} \\times 3x \\times \\frac{1}{2}x = 75$$","highlights":[{"text":"The area of the triangle is 75cm2.","location":"specification"}]},{"type":"text","order":3,"title":"Simplify and solve for x","content":"Let's simplify our equation:\n\n$$\\begin{aligned}\n\\frac{3}{4}x^2 &= 75 \\\\\n\\\\\nx^2 &= 75 \\times \\frac{4}{3} \\\\\n\\\\\nx^2 &= 100 \\\\\n\\\\\nx &= \\sqrt{100} \\\\\n\\\\\nx &= 10\n\\end{aligned}$$\n\n*(Note: Since $x$ represents a physical length, we only take the positive square root).* \n\nThus, $x = 10$."}]}],"generatedAt":"2026-06-13T03:12:04.756Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":50},"walkthroughVideo":null},{"id":"c6a79040-7c95-453e-9ae4-6581d1a522bd","revisionId":"430cabe3-425e-44a2-b5a1-021bf0b6b39a","questionNumber":null,"questionType":"LA","specification":"![A triangular garden plot has two sides of length (x + 3) metres and (x + 5) metres, with an included angle of 30 degrees.](https://assets.mathsgenie.co.uk/questions/7140fc6c-3b87-4837-b129-1edefa0b00e3.png)","difficulty":4,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":"calc","isCalculation":true,"parts":[{"id":"83df2179-92fe-4c4e-80c0-1743c2e215c4","label":"a","content":"The area of the garden plot is $15 \\text{ m}^2$. \nWork out the value of $x$. \nGive your answer to 3 significant figures.","markscheme":"- Sets up equation: $\\frac{1}{2}(x+3)(x+5)\\sin(30) = 15$ M1\n- Simplifies to $x^2 + 8x + 15 = 60$ or equivalent M1\n- Forms quadratic $x^2 + 8x - 45 = 0$ M1\n- Applies quadratic formula $x = \\frac{-8 \\pm \\sqrt{8^2 - 4(1)(-45)}}{2}$ M1\n- Calculates $x = 3.81$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up the area equation using the formula 1/2 ab sin C.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the equation to remove fractions and trig terms.","code":"M"},{"id":"p3","marks":1,"description":"Rearranges the simplified equation into a standard quadratic form (ax^2 + bx + c = 0).","code":"M"},{"id":"p4"

Finding the Area of Any Triangle

Question 5

Finding the Area of Any Triangle Questions

Finding the Area of Any Triangle

Question 5

Finding the Area of Any Triangle Questions