In circle A, ∠BAE ≅ ∠DAE. Circle A is shown. Line segments A B, A E, and A D are radii. Lines are drawn from point B to point E and from point E to point D to form secants B E and E D. Angles B A E and E A D are congruent. The length of B E is 3 x minus 24 and the length of E D is x + 10. What is the length of BE? 14 units 17 units 27 units 34 units

The missing figure is attached down

The length of BE is 27 units ⇒ 3rd answer

Step-by-step explanation:

In circle A:

We need to find the length of BE

∵ AB and AD are radii in circle A

∴ AB ≅ AD

In Δs EAB and EAD

∵ ∠BAE ≅ ∠DAE ⇒ given

∵ AB = AD ⇒ proved

∵ EA = EA ⇒ common side in the two triangles

- Two triangles have two corresponding sides equal and the

   including angles between them are equal, then the two

   triangles are congruent by SAS postulate of congruence

∴ Δ EAB ≅ Δ EAD ⇒ SAS postulate of congruence

By using the result of congruence

∴ EB ≅ ED

∵ EB = 3 x - 24

∵ ED = x + 10

- Equate the two expressions to find x

∴ 3 x - 24 = x + 10

- Add 24 to both sides

∴ 3 x = x + 34

- Subtract x from both sides

∴ 2 x = 34

- Divide both sides by 2

∴ x = 17

Substitute the value of x in the expression of the length of BE to find its length

∵ BE = 3 x - 24

∵ x = 17

∴ BE = 3(17) - 24

∴ BE = 51 - 24

∴ BE = 27

The length of BE is 27 units

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