phimaths Start free

Engineering and Science Admissions Test

ESAT Maths 1 Graphs questions

3 original ESAT Maths 1 graphs questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: Perpendicular gradients multiply to −1

Question 1ESAT Maths 1Standard

The line LL passes through (−1,−3)(-1, -3) and is perpendicular to the line y=12x+5y = \dfrac{1}{2}x + 5. What is the area of the triangle formed by LL and the two coordinate axes?
  1. A174\dfrac{17}{4}
  2. B214\dfrac{21}{4}
  3. C254\dfrac{25}{4}
  4. D274\dfrac{27}{4}
  5. E294\dfrac{29}{4}
  6. F334\dfrac{33}{4}
  7. G374\dfrac{37}{4}
  8. H252\dfrac{25}{2}
Show the answer and worked solution

Answer: C · Method: Perpendicular gradients multiply to −1

  1. LL has gradient −1÷12=−2-1 \div \dfrac{1}{2} = -2, so LL: y+3=−2(x+1)y + 3 = -2(x + 1).
  2. It crosses the yy-axis at −5-5 and the xx-axis at −52-\dfrac{5}{2}.
  3. Area =12×52×5=254= \tfrac{1}{2} \times \dfrac{5}{2} \times 5 = \dfrac{25}{4}.

Why the tempting wrong answers are wrong

  • H: The area of a triangle is half base times height.

Try it interactively →

Question 2ESAT Maths 1Standard

The line LL passes through (4,2)(4, 2) and is perpendicular to the line y=−12x−3y = -\dfrac{1}{2}x - 3. What is the area of the triangle formed by LL and the two coordinate axes?
  1. A88
  2. B99
  3. C192\dfrac{19}{2}
  4. D1010
  5. E1111
  6. F1616
  7. G1818
Show the answer and worked solution

Answer: B · Method: Perpendicular gradients multiply to −1

  1. LL has gradient −1÷−12=2-1 \div -\dfrac{1}{2} = 2, so LL: y−2=2(x−4)y - 2 = 2(x - 4).
  2. It crosses the yy-axis at −6-6 and the xx-axis at 33.
  3. Area =12×3×6=9= \tfrac{1}{2} \times 3 \times 6 = 9.

Why the tempting wrong answers are wrong

  • F: Perpendicular means the gradient is the negative reciprocal, not the same gradient.
  • G: The area of a triangle is half base times height.

Try it interactively →

Question 3ESAT Maths 1Standard

The line LL passes through (3,2)(3, 2) and is perpendicular to the line y=−2x−2y = -2x - 2. What is the area of the triangle formed by LL and the two coordinate axes?
  1. A14\dfrac{1}{4}
  2. B12\dfrac{1}{2}
  3. C34\dfrac{3}{4}
  4. D54\dfrac{5}{4}
  5. E94\dfrac{9}{4}
  6. F1616
Show the answer and worked solution

Answer: A · Method: Perpendicular gradients multiply to −1

  1. LL has gradient −1÷−2=12-1 \div -2 = \dfrac{1}{2}, so LL: y−2=12(x−3)y - 2 = \dfrac{1}{2}(x - 3).
  2. It crosses the yy-axis at 12\dfrac{1}{2} and the xx-axis at −1-1.
  3. Area =12×1×12=14= \tfrac{1}{2} \times 1 \times \dfrac{1}{2} = \dfrac{1}{4}.

Why the tempting wrong answers are wrong

  • B: The area of a triangle is half base times height.
  • F: Perpendicular means the gradient is the negative reciprocal, not the same gradient.

Try it interactively →

There are 20 graphs questions on phimaths.

Free, no sign-up. Get one wrong and you get a new question that needs the same method.

Start practising →

More ESAT Maths 1 topics