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Finding a triangle’s altitude from its sides

Lessons 31 · 1 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
31

Finding a triangle’s altitude from its sides

Textbook: pp. 101–102
GoalKnow how to compute the altitude of a triangle from its three sides with a formula.
New words
altitude · balandliksemi-perimeter · yarim perimetrfrom the sides · tomonlari bo‘yicharoot · ildiz
Explanation

Let the sides of a triangle be a, b, c and its semi-perimeter p = (a + b + c) : 2. The altitude h_c from vertex C to side c is found by applying the Pythagorean theorem to the two right triangles it makes. The result is the formula h_c = 2 : c · √(p(p − a)(p − b)(p − c)). The other altitudes are found the same way: use 2 : a for h_a and 2 : b for h_b. When computing, first find p, then the product under the root. The altitude to the longer side is smaller, and the altitude to the shorter side is larger.

Worked examples
Sides 13, 20, 21: p = 27 and 27 · 14 · 7 · 6 = 15 876 = 126². The altitude to 21 is 2 · 126 : 21 = 12.
Sides 13, 13, 24: p = 25 and 25 · 12 · 12 · 1 = 3600 = 60². The altitude to 24 is 2 · 60 : 24 = 5.
Class activity

“Compare the altitudes”: for the triangle 13, 20, 21 compute all three altitudes (approximately when they are fractions) and compare them with the sides.

Practice
1
What is the semi-perimeter of the triangle with sides 13, 20, 21?
2
In the triangle 5, 5, 6, p = 8. What is p(p − a)(p − b)(p − c)?
3
What is the altitude to the side 14 in the triangle 25, 25, 14?
4
Why is the altitude to the longest side of a triangle the smallest?