Question

In trapezoid ABCD AC is a diagonal and ∠ABC ≅ ∠ACD. Find AC if the lengths of the bases BC and AD are 14m and 28m respectively.

Answer 1

To find the length of AC, the diagonal in trapezoid ABCD, we can use the fact that ∠ABC ≅ ∠ACD and the lengths of the bases BC and AD. By setting up a proportion using the corresponding sides of the similar triangles ABC and ACD, we can find that AB/AC = 1/2. Multiplying the length of AB by 2 gives us AC = 28m.

Step-by-step explanation:

Given that ∠ABC ≅ ∠ACD and the lengths of the bases BC and AD are 14m and 28m respectively, we can find the length of AC, which is a diagonal in trapezoid ABCD.

1. Because the two angles are congruent, triangle ABC is similar to triangle ACD. This means their corresponding sides are proportional.
2. We can set up a proportion using the corresponding sides of the triangles:
   
   BC/AD = AB/AC
3. Substituting the values, we get 14/28 = AB/AC. Simplifying this equation gives us AB/AC = 1/2.
4. Since the lengths of AB and AC have a ratio of 1/2, we can determine AC by multiplying the length of AB by 2.
5. Therefore, AC = 2 * AB = 2 * 14 = 28m.

Answer 2

`AC=14√(2)m`

Step-by-step explanation:

we are given trapezoid ABCD

AC is a diagonal

∠ABC ≅ ∠ACD

we know that

BC and AD are parallel

so, <DAC = <ACB

so, ΔABC ≅ΔACD

Since, both triangles are similar

so, their ratios must be equal

we get

`(BC)/(AC) =(AC)/(AD)`

now, we can plug values

`(14)/(AC) =(AC)/(28)`

`AC^2=14* 28`

`AC^2=14^2* 2`

`AC=14√(2)m`