Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?

Question

Homi · Accepted Answer

We can construct a triangle with sides in a given ratio, but triangles with the same side ratio are not always congruent.

Step 1 — Constructing a triangle

Let us choose simple side lengths for our triangle. The ratio is 3 : 4 : 5. We can pick the actual side lengths as 3 cm, 4 cm, and 5 cm. This is a right-angled triangle, as $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ . Let us draw the longest side first.

Draw a line segment AB of length 5 cm.
Place the compass at point A.
Open the compass to 3 cm.
Draw an arc above the line segment AB.
Place the compass at point B.
Open the compass to 4 cm.
Draw another arc above AB.
The point where the two arcs intersect is point C.
Join A to C and B to C.
Triangle ABC is the required triangle.

Diagram 1

Step 2 — Congruence of triangles

Congruent triangles are triangles that have the exact same shape and the exact same size. Let us consider different triangles with the side ratio 3 : 4 : 5. We can multiply the ratio by a common factor, let's call it $k$ . If $k = 1$ , the sides are 3 cm, 4 cm, 5 cm. If $k = 2$ , the sides are $3 \times 2 = 6$ cm, $4 \times 2 = 8$ cm, $5 \times 2 = 10$ cm. If $k = 3$ , the sides are $3 \times 3 = 9$ cm, $4 \times 3 = 12$ cm, $5 \times 3 = 15$ cm.

These triangles all have the same shape. They are all right-angled triangles. However, their actual sizes are different. A triangle with sides 3, 4, 5 is smaller than one with sides 6, 8, 10. So, they are not the same size. Therefore, these triangles are not congruent. They are similar triangles, meaning they have the same shape.

Answer

(i) We can construct triangles with sides in the ratio 3 : 4 : 5. (ii) They will not be congruent to each other. (iii) Congruent triangles must have the same shape and size. Triangles with the same side ratio but different actual side lengths (e.g., 3,4,5 cm vs. 6,8,10 cm) have the same shape but different sizes. So, they are not congruent.

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