Final answer:
The height of triangle ABE is calculated to be 10 cm by establishing a proportion between the sides of the given triangle and a hypothetical similar smaller triangle, using the known lengths of the triangle's sides and the properties of similar triangles.
Step-by-step explanation:
To calculate the height, (h) cm, of triangle ABE using similar triangles, we need to first identify the known measurements: AB = 36 cm, BC = 10 cm, and AE = 20 cm. By the properties of similar triangles, the ratio of corresponding sides is constant. Therefore, if there exists a similar triangle A'B'C' with the same shape as ABE but scaled down, we can say that the ratio of AE to AB (the heights in triangle ABE) is the same as the ratio of A'C' to A'B' (the corresponding heights in triangle A'B'C').
In this case, we don't have concrete measurements for the smaller triangle, but we can assume that there is a hypothetical similar triangle where A'C' is half the length of AE, which would be 10 cm (considering the ratio provided: h₁ = 20 m, h₂ = 10 m). Using this hypothetical ratio, A'B' would then also be half the length of AB, which would be 18 cm.
Now that we have our proportional heights and bases for the triangles, we can set up a proportion:
AE / AB = A'C' / A'B'. Substituting the known values into this proportion, we get 20 cm / 36 cm = h / 18 cm. Cross-multiplying to solve for h gives us 20 cm × 18 cm = 36 cm × h, which simplifies to 360 cm² = 36h. Dividing both sides by 36 gives us the height h = 10 cm.