Ten seconds of drawing buys minutes of clarity. A rough figure shows structure the words hide — and quietly executes half the choices.
The problem
Points A(0, 0), B(6, 0) and C(0, 8) form a triangle.
How far is B from C?
Sketch the three points and the structure appears instantly: a right triangle with legs 6 and 8 along the axes — a 3-4-5 doubled, so BC = 10. B adds the legs, C multiplies them; both dissolve the moment the picture exists.
1
Know the drawing that fits each problem.
A 5 m ladder leans against a wall with its foot 3 m out. How high does it reach?
wall, ground, ladder
draw it: a right triangle, right angle at the foot of the wall
ladder = 5 is the hypotenuse
the side opposite the right angle
3² + h² = 5²
ground 3 is a leg, height h is the other
h² = 25 − 9 = 16, h = 4 m
a 3-4-5 triangle, seen only once it is drawn
Every problem type has a drawing like this one:
Problem smells like…
Draw a…
geometry with no figure given
rough figure, labelled with every number
coordinates
quick axes — even five gridlines is enough
inequalities, absolute value
number line with the cut points marked
ages, schedules, motion
timeline or a two-row table
2
Rough is enough — but keep proportions honest.
A at the origin, B on the x-axis, C on the y-axis
plot the three points
∠A = 90°
the axes meet at a right angle
AB = 6, AC = 8
read straight off the axes
BC = 10
the hypotenuse of a 6-8-10 (3-4-5 doubled)
The sketch is not art; it is a thinking surface. Draw a "6" leg longer than
a side of 4, keep right angles looking right, and label everything. A
roughly-to-scale figure lets your eye estimate the answer — and an eye
estimate of "about 10" already convicts 48 and 7 of being wrong.
Points off the axes: the run is the difference of the x's, the rise the difference of the y's.
With no axes given, we place a figure on them ourselves: a 6 by 4 rectangle with one corner at (0, 0) has corners (6, 0), (6, 4) and (0, 4), and the midpoint of its top side is (3, 4) — so that corner is √3² + 4² = 5 from it.
3
The number line settles inequality questions on sight.
Which x satisfy |x − 2| < 3?
mark 2, step 3 both ways
the points −1 and 5 appear
−1 < x < 5
"within 3 of 2" — visible, not memorised
Drawn on a line, "distance from 2 is less than 3" stops being a formula with
cases and becomes a picture with an inside.
4
Use the sketch as a lie detector on the choices.
48 ✗
longer than the whole figure — the sides are 6 and 8
7 ✗
shorter than the 8 leg; the hypotenuse beats both legs
14 ✗
the bent path B → A → C; the straight line BC is shorter
10 ✓
between 8 and 14 — the only survivor
Adding the legs
With no picture, “B is 6 away and C is 8 away” turns into BC = 6 + 8 = 14. That
adds the two legs — the trip B → A → C, round the corner. The sketch shows BC
cutting straight across, so it must be shorter than 14: √6² + 8² = 10.
Before computing, hold each choice against the picture: a length longer
than the whole figure, an angle fatter than the drawing shows, an area
bigger than the bounding box — cross them out. On a four-choice exam this
regularly leaves two candidates before any algebra starts.
5
Directions: net them, then draw one triangle.
A cyclist rides 5 km north, then 9 km west, then 17 km south. How far is she from where she started?
north–south: 17 south − 5 north = 12 km south
opposite directions cancel
east–west: 9 km west
nothing to cancel
distance = √12² + 9² = √225 = 15 km
the straight line home
Distance ridden is not distance from home
5 + 9 + 17 = 31 km is the trip, and it is always among the choices. The question
asks for the straight gap, which the sketch shows is far shorter.
6
Variations: the outside of a number line, and a sketch with two answers.
“At least” instead of “less than”: which x satisfy |x + 1| ≥ 4?
|x + 1| = |x − (−1)|: mark −1
the centre is −1, not 1
step 4 both ways: −5 and 3
the cut points
x ≤ −5 or x ≥ 3
at least 4 away: the two outer pieces
What is different: the answer is the outside of the picture, in two pieces, and the cut points are included because of the “=”.
Not −5 ≤ x ≤ 3
That is the inside — the numbers within 4 of −1. Test x = 0: |0 + 1| = 1, which is not
at least 4, so 0 cannot be in the answer.
Two pictures fit: A, B and C lie on a line with AB = 4 and BC = 7. How far apart can A and C be?
A — B — C: AC = 4 + 7 = 11
C beyond B, away from A
C — A — B: AC = 7 − 4 = 3
C back past A
What is different: the words do not fix the order of the points. Sketch every arrangement the wording allows — a “could be” question usually lists only one of the answers.
7
Four more sketches that replace a formula.
An isosceles triangle has equal sides 13 cm and base 10 cm. Find its area.
draw the height from the apex to the base
it lands at the midpoint
half-base 5, side 13: height = √169 − 25 = 12
one of two equal right triangles
area = (12)(10)(12) = 60 cm²
the whole base, not the half
On a number line, A is at −4 and B is at 11. Point P is on AB with AP : PB = 2 : 3. Where is P?
AB = 11 − (−4) = 15, split into 2 + 3 = 5 parts
each part is 3
AP = 2 parts = 6
measured from A
P = −4 + 6 = 2
a position, not a length
A lawn 10 m by 6 m has a path 1 m wide all round its outside. Find the area of the path.
outer rectangle: (10 + 2) by (6 + 2) = 12 by 8
the path adds its width on BOTH sides
path = 12 × 8 − 10 × 6 = 96 − 60 = 36 m²
outer minus inner — the corner squares are included
A jeep leaves a town at 6:00 a.m. at 40 km/h. A van leaves the same town at 7:00 a.m. on the same road at 50 km/h. When does the van catch up?
timeline: at 7:00 the jeep is already 40 km ahead
its one-hour head start
the van gains 50 − 40 = 10 km each hour: 40 ÷ 10 = 4 h
counted from 7:00, when the van starts
7:00 a.m. + 4 h = 11:00 a.m.
not 6:00 + 4 h
Where these go wrong
Using the slant side 13 as the height gives 65; forgetting the half gives 120. On the
line, reading the ratio from B gives 5, and stopping at the length 6 answers the wrong
question. A ten-second sketch shows P must sit nearer A.
On the exam
Many UPCAT-type items give no figure on purpose — ladders, walks in compass
directions, gardens with paths, points on a line. Sketch first, label every number,
and look for the right triangle; then hold each choice against the picture before
doing any algebra.
eye test10 is a bit longer than the 8 legexactly how the hypotenuse looks ✓
Remember the flow
Pick the right drawing · Label every number · Eye-test the choices
Ten seconds of pencil beats two minutes of confusion.
Now you try
Practice problem
The problem
x is negative and |x| > 3. Which of these could x be?
Sketch the line: "negative" keeps the left half, "|x| > 3" keeps what lies more than 3 from zero — the surviving region is left of −3, and only −5 lives there. B is negative but too close to zero; the number line shows that instantly.
