40
Properties of a right triangle
Textbook: pp. 104–105
GoalKnow the properties of a right triangle.
New words
hypotenuse · gipotenuzaleg (cathetus) · katetacute angles · o‘tkir burchaklarmedian to the hypotenuse · gipotenuzaga tushirilgan mediana
Explanation
In a right triangle the two sides forming the right angle are the legs, and the third side, facing the right angle, is the hypotenuse. Property 1: the two acute angles of a right triangle add up to 90°. Property 2: the leg opposite a 30° angle equals half the hypotenuse; proof: attach a triangle equal to the given one to form an equilateral triangle with 60° angles, where AB = AD = 2AC. Converse: if a leg equals half the hypotenuse, the angle opposite it is 30°. Property 3: the median to the hypotenuse equals half the hypotenuse. These properties solve many problems quickly.
Worked examples
In a right triangle, if one acute angle is 37°, the other is 90° − 37° = 53°.
With ∠B = 30° and hypotenuse AB = 18 cm, the leg AC opposite ∠B is 9 cm.
Class activity
“Half or full”: cards give a hypotenuse and an angle; pupils quickly find the leg opposite 30°.
Practice
1
In a right triangle one acute angle is 28°. Find the other (degrees).
62
2
The leg opposite 30° is 7 cm. Find the hypotenuse (cm).
14
3
The hypotenuse is 26 cm. Find the median to it (cm).
13
4
Why do the acute angles of a right triangle sum to 90°?
The total is 180°, the right angle takes 90°, leaving 180° − 90° = 90°.