Enlargement and Area: Why You Multiply by k², Not k

The question

In the diagram, triangle $PST$ is the image of triangle $PQR$ under an enlargement with centre $P$ .

The point $Q$ lies on $PS$ and the point $R$ lies on $PT$ . $PQ = 4\ \text{cm}$ and $QS = 6\ \text{cm}$ .

The area of triangle $PQR$ is $10\ \text{cm}^2$ .

Triangle PQR inside a larger triangle PST, both sharing vertex P. Q lies on PS with PQ = 4 cm and QS = 6 cm; R lies on PT. Triangle PQR has area 10 cm squared, and the quadrilateral QSTR between the two triangles is shaded. — Triangle PQR is enlarged about P to give triangle PST. The region QSTR is shaded.

(a) Find the scale factor of the enlargement. [1 mark]

(b) Calculate the area of the shaded region $QSTR$ . [3 marks]

Cambridge IGCSE note: using the relationship between the lengths and areas of similar shapes is Extended content in 0580.

The common wrong answer

Most students get part (a) right: the scale factor is $\dfrac{10}{4} = 2.5$ . Part (b) is where the marks go. The working I see most often is:

“The triangle is 2.5 times bigger, so its area is 2.5 times bigger.”

\text{Area of } PST = 2.5 \times 10 = 25\ \text{cm}^2

\text{Shaded area} = 25 - 10 = 15\ \text{cm}^2

The mistake is the first line. A scale factor of $2.5$ multiplies every length by $2.5$ . Area is length times length, so it is multiplied by $2.5$ twice. Multiplying the area by $k$ instead of $k^2$ is one of the errors Cambridge examiners point out on enlargement questions: candidates enlarge an area as if it were a length.

The correct working

(a) The scale factor compares distances from the centre $P$ . First find $PS$ :

PS = PQ + QS = 4 + 6 = 10\ \text{cm}

k = \frac{PS}{PQ} = \frac{10}{4} = 2.5

The scale factor is $2.5$ (or $\dfrac{5}{2}$ ).

(b) Under an enlargement with scale factor $k$ , the area of the image is $k^2$ times the area of the object:

k^2 = 2.5^2 = 6.25

\text{Area of } PST = 6.25 \times 10 = 62.5\ \text{cm}^2

The shaded region is the big triangle with the small triangle taken out:

\text{Shaded area} = 62.5 - 10 = 52.5\ \text{cm}^2

The area of the shaded region $QSTR$ is $52.5\ \text{cm}^2$ .

Why it works

Think of the area of a triangle as $\dfrac{1}{2} \times \text{base} \times \text{height}$ . An enlargement with scale factor $2.5$ makes the base $2.5$ times longer and the height $2.5$ times longer. So the area becomes

\frac{1}{2} \times (2.5 \times \text{base}) \times (2.5 \times \text{height}) = 2.5^2 \times \left(\frac{1}{2} \times \text{base} \times \text{height}\right)

which is $6.25$ times the original area. The same is true for any shape, not just triangles: lengths scale by $k$ , areas by $k^2$ (and, for solids, volumes by $k^3$ ).

You can sanity-check it with a simple case. Enlarge a $1\ \text{cm}$ by $1\ \text{cm}$ square with scale factor $2$ . You get a $2\ \text{cm}$ by $2\ \text{cm}$ square, which holds four of the small squares, not two.

A quick check for this question: the shaded region should be $k^2 - 1 = 5.25$ times the small triangle, and $5.25 \times 10 = 52.5$ . It matches.

How to spot it next time

Before you use the scale factor, ask what you are scaling. Lengths use $k$ ; areas use $k^2$ .
Measure the scale factor from the centre of enlargement: use $PS$ and $PQ$ , not $QS$ and $PQ$ . Using $\dfrac{6}{4}$ is a second trap in this diagram.
When a question asks for a shaded region between an object and its image, find the image area first, then subtract the object area.