13
Practical exercise and application
Textbook: pp. 42–43
GoalPractical uses of similarity: measuring heights from shadows and distances, and solving problems about the diagonals of a trapezoid.
New words
indirect measurement · masofani bilvosita o‘lchashdiagonals of a trapezoid · trapetsiya diagonallaribases · asoslarscale · masshtab
Explanation
The height of a tree or building, or the width of a river, that cannot be measured directly can be found with similar triangles: for simultaneous shadows the ratio of heights equals the ratio of shadow lengths. In a trapezoid the diagonals form two similar triangles at their intersection: the triangles resting on the bases are similar with a coefficient equal to the ratio of the bases. So the pieces of the diagonals are proportional to the bases. Similarity also appears as scale on maps and drawings. In a problem it is important to sketch a simple figure and name the similar pair clearly.
Worked examples
A trapezoid has bases BC = 4 and AD = 10 and diagonal BD = 21. △BOC ∽ △DOA, so BO : OD = 4 : 10 = 2 : 5. Hence BO = 21 · 2/7 = 6 and OD = 15.
Finding a river’s width: two similar right triangles are laid out on the bank. The small one has legs 4 m and 6 m; in the large one the leg matching 4 m is 20 m, so the width (the leg matching 6 m) is 6 · 20/4 = 30 m. Measure only with adults and away from the water’s edge.
Class activity
“Schoolyard surveyor”: groups measure the height of a flagpole or tree by the shadow method, compare results and discuss sources of error.
Practice
1
A person 160 cm tall casts a 120 cm shadow. At the same time a pole’s shadow is 450 cm. How tall is the pole (cm)?
600
2
A trapezoid has bases 6 and 9 and a diagonal of 25 cm. What parts does the intersection point cut the diagonal into?
10 cm and 15 cm
3
The similar triangles formed by a trapezoid's diagonals have k = 2/3 and the smaller has area 12. What is the area of the larger?
27
4
Why does the shadow method give a correct result only when both shadows are measured at the same time?
Only if the sun's angle is the same are the triangles' angles equal and the triangles similar.