Similar Triangles: Proportional Sides

Two triangles are similar if they have the same shape but possibly different sizes. This means:

1. All three corresponding angles are equal.
2. All three pairs of corresponding sides are in the same ratio.

The ratio of corresponding sides is called the scale factor $k$. If triangle 2 is twice the size of triangle 1, then $k = 2$ and every side of triangle 2 is twice the corresponding side of triangle 1.

Step-by-Step Solution

Given: Triangle 1 has sides $a_1 = 3$, $b_1 = 4$, $c_1 = 5$. Triangle 2 is similar to it with $a_2 = 6$.

Find: The other sides $b_2$ and $c_2$, the scale factor, and the area ratio.

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Step 1 — Find the scale factor.

$k = \dfrac{a_2}{a_1} = \dfrac{6}{3} = 2$

Step 2 — Apply the scale factor to the other sides.

$b_2 = k \cdot b_1 = 2 \cdot 4 = 8$

$c_2 = k \cdot c_1 = 2 \cdot 5 = 10$

Step 3 — Verify the side ratios are all equal.

$\dfrac{a_2}{a_1} = \dfrac{6}{3} = 2$

$\dfrac{b_2}{b_1} = \dfrac{8}{4} = 2$

$\dfrac{c_2}{c_1} = \dfrac{10}{5} = 2$

All three ratios equal 2 — that's exactly what similarity means.

Step 4 — Compute the area ratio.

Triangle 1 (a 3-4-5 right triangle) has area:

$A_1 = \tfrac{1}{2}(3)(4) = 6$

Triangle 2:

$A_2 = \tfrac{1}{2}(6)(8) = 24$

The ratio of areas:

$\dfrac{A_2}{A_1} = \dfrac{24}{6} = 4 = k^{2}$

Areas scale as the square of the scale factor. This is a general fact for any similar 2D shapes — doubling all linear dimensions quadruples the area.

Step 5 — Volumes scale as the cube.

If you had similar 3D objects with scale factor $k$, the volumes would scale as $k^{3}$. Doubling the linear size of an object multiplies its volume by 8. This is why ants are so disproportionately strong: they don't scale up like animals; if you doubled an ant's linear size, its weight (volume × density) would grow $8\times$ but its muscle cross-section (area) only $4\times$ — making it relatively weaker.

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Answer: Triangle 2 has sides $a_2 = 6$, $b_2 = 8$, $c_2 = 10$, with scale factor $k = 2$. Its area is 4 times triangle 1's, because $k^{2} = 4$.

In general: if two figures are similar with scale factor $k$, then lengths scale as $k$, areas scale as $k^{2}$, and volumes scale as $k^{3}$.

Try It

• Adjust the scale factor $k$ — see triangle 2 grow or shrink.
• The HUD shows both the side ratios (always equal to $k$) and the area ratio (always $k^{2}$).
• Try $k = 0.5$ — triangle 2 becomes half the size, with $1/4$ the area.