The bases of a trapezoid are 6'' and 9'' long. If the distance between the bases is 8'', find the distance from the intersection of the diagonals to the longer base.

Question

The bases of a trapezoid are 6'' and 9'' long. If the distance between the bases is 8'', find the distance from the intersection of the diagonals to the longer base.

Answer 1

To find the distance from the intersection of the diagonals to the longer base of a trapezoid, we can use the property that the diagonals of a trapezoid divide each other into segments that are proportional.

Let's label the trapezoid as ABCD, with bases AB and CD. The diagonals, AC and BD, intersect at a point O.

First, we need to find the length of the shorter base, CD. We have been given that the longer base, AB, is 9 inches, and the distance between the bases, the height of the trapezoid, is 8 inches.

Next, we can use the property of similar triangles to find the length of the other segment of the longer base. Let's consider triangle ABO and triangle BCO. They are similar because they share angle AOB and angle BOC.

The ratio between the sides of similar triangles is equal. Therefore, since AB is to BC as AO is to BO, we have:
AB/BC = AO/BO

Substituting the given values, we have:
9/BC = AO/BO

Now, we can substitute the value of AO by the sum of AO and BO, which is the longer base, AB:
9/BC = AB/BO

Rearranging the equation, we get:
BC/AB = BO/9

Solving for BC, we have:
BC = (BO/9) * AB

Now, let's apply the similar property to triangle ADC and triangle BDC. Since they share angle ADC and angle BDC, we have:
BC/CD = BO/BD

Substituting the values we know, we have:
[(BO/9) * AB] / CD = BO/BD

Rearranging the equation to isolate CD, we get:
CD = [(BO/BD) * 9 * AB] / BO

Next, we can use the property of similar triangles again, this time with triangle ADO and triangle CDO. They share angle AOD and angle COD. Therefore, we have:
AD/CD = AO/CO

Substituting the known values, we have:
AD/[(BO/BD) * 9 * AB / BO] = AO/CO

Simplifying the equation, we get:
AD = [(AO/CO) * 9 * AB] / (BO/BD)

Finally, to find the distance from the intersection of the diagonals to the longer base, we need to find the value of AD. Substitute the lengths of the given values and solve the equation to find the answer.