The legs of a right-angled triangle have lengths 3 and 6. Find: a) the hypotenuse;

The legs of a right-angled triangle have lengths 3 and 6. Find: a) the hypotenuse; b) the area of the triangle; c) the height lowered to the hypotenuse.

Let the legs of a right-angled triangle a and b, hypotenuse c: a = 3, b = 6.
a) Using the Pythagorean theorem, we find the hypotenuse:
c ^ 2 = a ^ 2 + b ^ 2;
c = √ (a ^ 2 + b ^ 2);
c = √ (3 ^ 2 + 6 ^ 2) = √ (9 + 36) = √45 = 3√5 (conventional units).
b) The area of ​​a right-angled triangle is equal to half the product of its two legs and is found by the formula:
S = ab / 2;
S = (3 * 6) / 2 = 18/2 = 9 (conventional square units).
c) The height h, lowered from a right angle to the hypotenuse in a right-angled triangle, is found by the formula:
h = ab / c;
h = (3 * 6) / 3√5 = 6 / √5.
Let’s get rid of irrationality in the denominator:
h = 6√5 / 5 conventional units.
Answer: a) c = 3√5 conventional units; b) S = 9 conventional square units; c) h = 6√5 / 5 conventional units.