33
Problem solving
Textbook: pp. 96–97
GoalSolve mixed problems on triangles and parallelograms (sum of squares of diagonals, median, height) with the trigonometric theorems.
New words
diagonals of a parallelogram · parallelogramm diagonallarimedian · medianaheight · balandliksum · yig‘indisi
Explanation
In a parallelogram the sum of the squares of the diagonals equals twice the sum of the squares of the sides: d₁² + d₂² = 2(a² + b²). It follows by applying the law of cosines to two adjacent triangles and cos(180° − α) = −cosα. If the area of a triangle is found with Heron's formula, the height to any side is h = 2S/a. The length of a median to side a is m = ½√(2b² + 2c² − a²). In a hard problem first find the area or an auxiliary side, then the required quantity. Check the result by substituting back.
Worked examples
A parallelogram has sides 5 and 7 and one diagonal 8: the square of the other is 2(25 + 49) − 64 = 84, so d₂ = 2√21.
In the triangle with sides 7, 15, 20 (S = 42 by Heron’s formula) the height to the side 7 is h = 2·42/7 = 12.
Class activity
“Choose the formula”: for each card problem, quickly say which formula (Heron, median, diagonals, cosines) fits, then solve.
Practice
1
In a triangle with sides 6, 8, 10 how long is the median to the side 10?
5
2
A parallelogram has sides 3 and 4. What is the sum of the squares of its diagonals?
50
3
In a triangle with sides 5, 5, 6, what is the height to the base 6?
4
4
Why is the median to the hypotenuse of a right triangle half of it?
By m = ½√(2a² + 2b² − c²) and c² = a² + b² we get m = c/2.