In the coordinate axis above, line segment AC is three times

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?


4

4\sqrt{2}

6

8

It cannot be determined from the information given.

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?

A. 4
B. \(4\sqrt{2}\)
C. 6
D. 8
E. It cannot be determined from the information given.

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?

(A) 4
(B) 4\sqrt{2}
(C) 6
(D) 8
(E) It cannot be determined from the information given.

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?

A. 4
B. \(4\sqrt{2}\)
C. 6
D. 8
E. It cannot be determined from the information given.

I'm happy to help.

This is tricky because the diagram is not to scale, but we can use proportional reasoning. If 3*AB = BC, that means (BC)/(AC) = 2/3, that means from the origin to B must be (2/3) of the distance from the x-axis up to point A --- (2/3) of 6 is 4, so B = (4, 0). That's a 45 degree line, so C = (4, 0), and D = (-4, 0). Here's a scaled diagram:

DOB is a 45-45-90 triangle, with DO = OB = 4, so area = 8. Answer = (D).

Does all this make sense?
Mike

hey,
a slight correction. You have calculated B to be (4,0). It should be (0,4) as it lies on y axis where X=0

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?

4
4\sqrt{2}
6
8

It cannot be determined from the information given.

You can use similar triangles concept here.
Drop a perpendicular AE on the y axis.



Now, triangle BEA is similar to triangle BOC (by the AA rule. Angle ABE = Angle CBO since they are vertically opposite and both triangles have right angles)
Since AB/BC = 1/2, so BE/BO = AE/CO = 1/2
Since AE = 2, CO = 4. Also since EO is 6, BE = 2 and BO = 4.
So BOC is isosceles right triangle so both angles OBC and OCB are 45. This means angle OBD is 45 (since angle DBC is 90) making triangle DOB isosceles as well. Since BO = 4, DO = 4 as well.

Area of triangle DOB = (1/2)*4*4 = 8

Hi Karishma:
Typically, the definition of similarity between two triangles is as follows:

Triangles are similar if:
AAA (angle angle angle)
All three pairs of corresponding angles are the same.
See Similar Triangles AAA.
SSS in same proportion (side side side)
All three pairs of corresponding sides are in the same proportion
See Similar Triangles SSS.
SAS (side angle side)
Two pairs of sides in the same proportion and the included angle equal.
See Similar Triangles SAS.

Here is the source of information:
https://www.mathopenref.com/similartriangles.html

My Question:
Instead of 3 angles (AAA), can we just consider two angles (AA) to conclude that two triangles are similar?
Please share your thoughts.
Thanks in advance.

Regards,
Yosita

You can use similar triangles concept here.
Drop a perpendicular AE on the y axis.



Now, triangle BEA is similar to triangle BOC (by the AA rule. Angle ABE = Angle CBO since they are vertically opposite and both triangles have right angles)
Since AB/BC = 1/2, so BE/BO = AE/CO = 1/2
Since AE = 2, CO = 4. Also since EO is 6, BE = 2 and BO = 4.
So BOC is isosceles right triangle so both angles OBC and OCB are 45. This means angle OBD is 45 (since angle DBC is 90) making triangle DOB isosceles as well. Since BO = 4, DO = 4 as well.

Area of triangle DOB = (1/2)*4*4 = 8

In the coordinate axis above, line segment AC is three times as long as line segment AB. In addition, segment AC is perpendicular to segment BD. What is the area of triangle DOB?

A. 4
B. \(4\sqrt{2}\)
C. 6
D. 8
E. It cannot be determined from the information given.