Asad Abbas said:
I need some help in my maths homework.In the question the kite is divided into four triangles.I also need help in another question but it was to complicated to explain so I was trying to attach an attachment but I couldn't.I was hoping if you could help me.
Q.ABCD is a kite.AC=10cm BD=12cm.AC is a horizontal line and BD is a vertical line.BD is also the axis of symmentry.Find the area.
Did you draw a sketch? If you do, you will see that the diagonals of the kite intersect at a point I'll call E, creating four small triangles....triangle AEB, triangle CEB, triangle DEC and triangle DEA.
Because BD is a line of symmetry, the four angles at E are right angles, meaning those four small triangles are all right triangles.
Remember that the area of a right triangle is (1/2)*(product of the legs). Applying this to the small triangles, we see that
area triangle AEB = (1/2)*AE*BE
area triangle CEB = (1/2)*CE*BE
area triangle CED = (1/2)*CE*DE
area triangle DEA = (1/2)*AE*DE
The area of the kite is the sum of the areas of all four of these triangles, so
area kite ABCD = (1/2)*AE*BE + (1/2)*CE*BE + (1/2)*CE*DE + (1/2)*AE*DE
The first two terms on the right side have a common factor of (1/2)*BE, and the last two have a common factor of (1/2)*DE. Let's remove the common factor from each pair of terms:
area kite ABCD = (1/2)*BE(AE + CE) + (1/2)*DE(CE + AE)
AND, AE + CE = AC.....
area kite ABCD = (1/2)*BE*AC + (1/2)*DE*AC
Now, (1/2)*AC is a common factor in both terms of the right side....and we have
area kite ABCD = (1/2)*AC*(BE + DE)
Since BE + DE = BD, we can make a substitution.....
area kite ABCD = (1/2)*AC*BD
I know...you are saying "Wait....that's a WHOLE LOT of work!" Well, yes, it was. But look what we have proved!
AC and BD are the two diagonals of a kite....and the
area of that kite turns out to be (1/2)*(sum of the diagonals)!
You may have proved this already...but did not realize you could apply it to any problem where you know the diagonals of a kite, and need the area of the kite.
It should be easy for you to find the area of kite ABCD now, I hope.