Question

In $\triangle abc,$ we know the side lengths $ab=9\sqrt 2$, $bc=10\sqrt 2$, $ca=11\sqrt 2$. find the height of $\triangle abc$ from $a$ to $bc.$

Answer 1

To find the height of triangle ABC from point A to BC, use the Pythagorean theorem: square the hypotenuse and subtract the squares of the other two sides. Then, take the square root to find the height.

Step-by-step explanation:

To find the height of triangle ABC from point A to BC, we need to draw an altitude from point A to BC. This altitude will be perpendicular to BC and will intersect BC at a right angle. Let's label the height as 'h'. According to the Pythagorean theorem, the square of the height is equal to the difference between the square of the hypotenuse and the sum of the squares of the other two sides.

So, using the given side lengths:

AB = 9√2, BC = 10√2, and CA = 11√2

We have:

h^2 = AB^2 - (BC^2 - CA^2)

h^2 = (9√2)^2 - ((10√2)^2 - (11√2)^2)

h^2 = 162 - (200-242)

h^2 = 162 - 42

h^2 = 120

Taking the square root of both sides, we get:

h = √120