Call the intersection point of the two diagonals X.
Looking at Triangle BCX: ∠BCX = 85°, BX = 10 and CX = 7.
Law of Sines for Triangle BCX: sin(∠BCX) / BX = sin(∠CBX) / CX
---> sin(85°) / 10 = sin(∠CBX) / 7
---> 7 x sin(85°) / 10 = sin(∠CBX)
1) Find ∠CBX.
2) Use ∠CBX and ∠BCX to find ∠BXC.
3) Use ∠BXC to find ∠BXA.
4) Use the Law of Cosines on Triangle AXB to find AB.
5) Use the Law of Sines on Triangle AXB to find ∠ABX.
6) Find ∠ABC.
7) Find ∠BAD.
Draw a picture of this rectangle with its diagonals.
This picture contains four triangles. You know that two sides of each of these interior triangles are each 6 cm long. In two of the triangles, the angle contained between these sides is 30° and in the other two triangles, the angle is 150°.
One formula for the area of a triangle is: Area = ½ · a · b · sin(C) Where a and b are the lengths of the two sides that contain ∠C.
So the area of two of the triangles is each: Area = ½ · 6 · 6 · sin(30°)
while the area of each of the other two triangles is: Area = ½ · 6 · 6 · sin(150°)
Add these four triangular areas to get the area of the rectangle.
Start by drawing a picture of a rhomus with its diagonals.
The diagonals of that rhombus are perpendicular to each other; therefore, you have a picture that contains 4 congruent right triangles.
Since the diagonals are 6 cm and 8 cm, the sides of each of the right triangles are 3 cm and 4 cm. Calculate the hypotenuses of the right triangles (they are all the same size) and add these four sides together to get the perimeter of the rhombus.
Since π is an irrational number, it will remain an irrational number if it is multiplied by a non-zero integer (8) and divided by an integer (3).
A rational number is a number that is (or can be placed into) a fraction of integers (but the denominator cannot be zero). Although 8π/3 is a fraction, it is not a fraction of integers.
@@ End of Day Wrap Mon 29/12/14 /12/14 Sydney, Australia Time 11:20pm ♪ ♫
There were a number of good questions today. Our industrious and learned answerers were CPhill, Rosala, Alan, Juhaszbenedek08, Tiagunix and geno3141. Thanks all.
Now for some interest posts:
The first four were all suggested by CPhill. I am always pleased when people request interest posts to be included. Thanks Chris
1) Simplifying indices - A pretty cool question
2) An interesting co-ordinate geometry question. Thanks to Melody and Chris
3) An interesting trapezium area question. Thanks Chris
4) Finding the 4th side of a quadrilateral that is circumscribed by a circle. Thanks Melody and Chris
5) finding root using secant method. Thanks Alan
And that is it for today
♫♪ ♪ ♫ ♬ ♬ MELODY ♬ ♬ ♫♪ ♪ ♫
1) How do you change different base numbers to base 10 ? In this case base 2. Thanks Gino
2) A simple but important concept for our younger followers Thanks Geno
3) Geno answered lots of great geometry questions - most related to rhombuses. This one didn't
4) What is Phi ( φ )? Thanks Chris
♫♪ ♪ ♫ ♬ ♬ MELODY ♬ ♬ ♫♪ ♪ ♫