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Q: Find the scale factor Abe to acd?
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How many 3 letter combination are possible from the first five letters of the laphabet?

10 (x 6 if no repetition is allowed but abc is not regarded as the same as acb), 125 if it is. ABC/ABD/ABE/ACD/ACE/ADE/BCD/BCE/BDE/CDE


How many triangles can be formed using six points on a plane no three of which are on the same straight line?

20: abc, abd, abe, abf, acd, ace, acf, ade, adf, aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf, cef, def.


How many combinations can be made with a b c d e and f if 3 letters taken at a time?

If you can only take one of each letter, it would be 19: abc, abd, abe, abf, acd, ace,acf,ade,adf,aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf ,def.


Diagonals ac and bd of a trapezium abcd with ab parallel to CD intersect each other at o.prove that area of triangle aod equals area of boc?

First consider triangles ACD and BCD. They share a common base, CD, and their height is the distance between the parallel lines AB and CD. Consequently, area(ACD) = area(BCD) . . . . . . . . eqn 1 But ACD = AOD + OCD and BCD = BOC + OCD So area(ACD) = area(AOD) + area(OCD) and area(BCD) = area(BOC) + area(OCD) Substituting in eqn 1, area(AOD) + area(OCD) = area(BOC) + area(OCD) area(AOD) = area(BOC)


What is the proof that equiangular triangle is also called equilateral triangle?

Label the triangle ABC. Draw the bisector of angle A to meet BC at D. Then in triangles ABD and ACD, angle ABD = angle ACD (equiangular triangle) angle BAD = angle CAD (AD is angle bisector) so angle ADB = angle ACD (third angle of triangles). Also AD is common. So, by ASA, triangle ABD is congruent to triangle ACD and therefore AB = AC. By drawing the bisector of angle B, it can be shown that AB = BC. Therefore, AB = BC = AC ie the triangle is equilateral.