In a triangle ABC,AB=5cm,BC=12cm,angle ABC=90°calculate 1)the length of AC. 2)the area of the triangle 3)the length of the perpendicular from B to AC
c^2 = a^2+b^2
a = 1/2 bh
If the perp from B to AC is BD, then using similar triangles,
BD/AB = BC/AC
To answer the questions, we can use the Pythagorean theorem, the formula for the area of a triangle, and the concept of similar triangles.
1) To find the length of AC, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC).
So, in this case, we have:
AC^2 = AB^2 + BC^2
AC^2 = 5^2 + 12^2
AC^2 = 25 + 144
AC^2 = 169
Taking the square root of both sides, we get:
AC = √169
AC = 13 cm
Therefore, the length of AC is 13 cm.
2) To find the area of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height.
In this case, AB can be considered as the base, and the height can be the distance from C to AB. To find the height, we can use the fact that the perpendicular from the right angle (B) to the hypotenuse (AC) creates similar triangles.
Since the triangle is a right triangle, the perpendicular from B to AC will be the height of the triangle. Thus, the length of the perpendicular from B to AC is equal to the length of the height.
To calculate the length of the perpendicular from B to AC, we can use the fact that the triangles ABC and BDC are similar triangles.
Using the property of similar triangles, we can set up the following proportion:
AB/BC = BC/DC
Substituting the known values:
5/12 = 12/DC
Cross-multiplying:
5*DC = 12*12
5*DC = 144
DC = 144/5
DC = 28.8 cm
So, the length of the perpendicular from B to AC (which is the height of the triangle) is 28.8 cm.
Now, using the formula for the area of a triangle:
A = (1/2) * AB * height
A = (1/2) * 5 * 28.8
A = 72 square cm
Therefore, the area of the triangle is 72 square cm.
To summarize:
1) The length of AC is 13 cm.
2) The area of the triangle is 72 square cm.
3) The length of the perpendicular from B to AC (height of the triangle) is 28.8 cm.