Trapezoid

Draw line BH vertically intersect CD at point H. Since height h is 4, which means the length of BH = 4. And by the graph above, we can get angle BCD is 60°, angle HBC is 30°. Therefore, length CH is half of BC. On the basis of Using Pythagorean theorem we deduce that length BC = 8/(V3) and length CH = 4/(V3)
The trapezoid area formula is (AB + CD) * BH / 2.
By (1) alone, length AB is available, height BH is available as well, but not length CD. Thus,
(1) alone is NOT sufficient.
By (2) alone, we can deduce that it’s a isosceles trapezoid since angle ADC = BCH = 60°, but have no idea of AB.
Finally, with (1) and (2) together, AB = 5, CD = 2*(CH)+AB = 8/(V3)+5, BH = 4, the area is
(5+5+8/(v3))*4/2 = 20+16/(V3). These 2 conditions together do make sense.
(!!!! Note that V means square root. V3 = square root of 3. !!!!)

Pls find details in attachment.
Hope it helps.

sayysong,

Thank you very much; this was really quality work.
surprised about the coincidence you just joined today !!

What is your location ?

heshamelaziry wrote:Could you guys tell me what is the function of angel 120 in this problem ? I think the problem can be solved without it ?

Hi heshamelaziry,

nope. Without the 120 angle we would have to just assume that it is an isoceles trapezoid, and we don't know if that is the case (even though, in the figure, it looks like an isoceles trapezoid).

Testluv wrote:

heshamelaziry wrote:Could you guys tell me what is the function of angel 120 in this problem ? I think the problem can be solved without it ?

Hi heshamelaziry,

nope. Without the 120 angle we would have to just assume that it is an isoceles trapezoid, and we don't know if that is the case (even though, in the figure, it looks like an isoceles trapezoid).

From info in the stem we know that angels B and C measure 120 and 60. why couldn't we say that angels A and D measure 120 and 60 respectively ? Is it because the figure is not drwan to scale? but if it is a trapezoid, angels A and D must be 120 and 60 ?

Please help

heshamelaziry wrote:

Testluv wrote:

heshamelaziry wrote:Could you guys tell me what is the function of angel 120 in this problem ? I think the problem can be solved without it ?

Hi heshamelaziry,

nope. Without the 120 angle we would have to just assume that it is an isoceles trapezoid, and we don't know if that is the case (even though, in the figure, it looks like an isoceles trapezoid).

From info in the stem we know that angels B and C measure 120 and 60. why couldn't we say that angels A and D measure 120 and 60 respectively ? Is it because the figure is not drwan to scale? but if it is a trapezoid, angels A and D must be 120 and 60 ?

Please help

Not quite. In an isoceles (ie, a pretty and symmetrical) trapezoid, the base angles are equal. If we knew that this was an isoceles trapezoid, then you would be correct. But you can have an assymetric trapezoid, in which case the angles can be different. Remember, the definition of trapezoid is just one pair of parallel lines.

Imagine that line DC (the base) extended farther to the right. Then, the "arm" BC will have to stretch out. The figure looks symmetrical; but that does not it mean that it is!

My conclusion: all Geometry figures in the test are not what they look like. I have to find evidence to prove that they are indeed what they look like.

heshamelaziry wrote:My conclusion: all Geometry figures in the test are not what they look like. I have to find evidence to prove that they are indeed what they look like.

Excellent conclusion!

Let me modify it just a tiny bit:

All geometry figures in the test are not NECESSARILY what they look like. You need to find evidence for whether or not they are, in fact, what they look like!