triangle ABC is right angled at A. AD is perpendicularto BC. If AB= 5cm,BC= 13cm and AC=12 cm.find the area of triangle ABC. Also find the length of AD.

The area is (1/2)*5*12 = 30 cm^2

You get that from the two perpendicular legs, which form a base and height.

Since AD is the height h' from base b' = BC,
(1/2)*h'*13 = 30
h' = 60/13 = 4.615 = length of AD

To find the area of triangle ABC, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the base of the triangle is BC and the height is AD.

First, let's find the length of AD. Since triangle ABC is right-angled at A, AD is the altitude or height of the triangle.

We can use the Pythagorean theorem to find the length of AD. According to the theorem, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC) in a right-angled triangle. Applying the theorem:

AC^2 = AB^2 + BC^2
12^2 = 5^2 + 13^2
144 = 25 + 169
144 = 194

This equation is not true, which means I made a mistake. Let's correct it.

AC^2 = AB^2 + BC^2
AC^2 = 5^2 + 13^2
AC^2 = 25 + 169
AC^2 = 194

Now, let's find the square root of both sides to get the length of AC:

AC = √194
AC ≈ 13.928 cm

Next, let's find the area of triangle ABC:

Area = (1/2) * BC * AD
Area = (1/2) * 13 * AD
Area = (13/2) * AD

To find the value of AD, we need to use the fact that AD is perpendicular to BC, which means it forms a right angle with BC.

Since ABC is a right-angled triangle and AD is perpendicular to BC, we can use similar triangles to find the length of AD.

In triangle ABC, we have the ratios:

AD/AB = BC/AC

Substituting the given values:

AD/5 = 13/12

Cross-multiplying:

12 * AD = 5 * 13
12 * AD = 65

Now, solving for AD:

AD = 65 / 12
AD ≈ 5.417 cm

Now that we have the length of AD, we can substitute it into the formula for the area of triangle ABC to find the answer:

Area = (13/2) * AD
Area = (13/2) * 5.417
Area ≈ 35.062 cm^2

Therefore, the area of triangle ABC is approximately 35.062 cm^2 and the length of AD is approximately 5.417 cm.