In right triangle ABC, M and N are midpoints of legs AB and BC, respectively. Leg AB is 6 units long, and leg BC is 8 units long. How many square units are in the area of triangle APC?
so, where is P?
intersection
of which lines?
I don't mind helping to find answers, but I hate having to provide the questions as well ...
To find the area of triangle APC, we need to first find the lengths of its sides. Let's start by finding the length of side AC.
Since M is the midpoint of AB, and AB is 6 units long, then AM is half of AB, which is 6/2 = 3 units long.
Similarly, since N is the midpoint of BC, and BC is 8 units long, then BN is half of BC, which is 8/2 = 4 units long.
Now, we can find the length of AC by using the Pythagorean theorem. In a right triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AM and BN).
AC^2 = AM^2 + BN^2
AC^2 = 3^2 + 4^2
AC^2 = 9 + 16
AC^2 = 25
Taking the square root of both sides gives us the exact length of AC:
AC = √25
AC = 5 units
Now that we know the lengths of the sides AP, PC, and AC, we can calculate the area of triangle APC using the formula for the area of a triangle:
Area = (base * height) / 2
In this case, the base of the triangle is AC and the height is the length of the altitude drawn from point P to side AC.
The altitude divides the triangle into two smaller triangles, so we need to find the length of this altitude.
Since P is not one of the midpoints, we can find the length of the altitude by using similar triangles. Triangle ANP and triangle CNP are similar to triangle ABC, so the ratios of their sides are equal.
The ratio of the sides of similar triangles is equal to the ratio of their corresponding altitudes.
In this case, the ratio of the sides is 3/6 = 1/2, and the corresponding altitude is the length of PN.
The length of PN is half of the length of BC, which is 8/2 = 4 units.
So, the height of triangle APC is 4 units.
Now we can find the area:
Area = (base * height) / 2
Area = (5 * 4) / 2
Area = 20 / 2
Area = 10 square units
Therefore, the area of triangle APC is 10 square units.