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Angle DAC= angle BAD
what is the length of side bd round to one decimal place

Angle DAC= angle BAD what is the length of side bd round to one decimal place-example-1

2 Answers

5 votes

The length of BD ≈ 3.6 cm.

The diagram shows a triangle ABC with sides AB = 8.1 cm, AC = 5.9 cm, and CD = 2 cm. We are asked to find the length of side BD and side AD, given that angles DAC and BAD are equal.

Solution:

1. Identify the relevant triangles: We can see two smaller triangles within the larger triangle ABC: triangle ADC and triangle ABD.

2. Apply the Law of Cosines to triangle ADC:

We know AC = 5.9 cm and CD = 2 cm, and we need to find AD. Angle DAC is given as equal to angle BAD, so we can use the Law of Cosines to find AD:


  • AD^2 = AC^2 + CD^2 - 2 * AC * CD * cos(DAC)

  • AD^2 = 5.9^2 + 2^2 - 2 * 5.9 * 2 *cos(90°) // Since angles DAC and BAD are equal, and triangle ADC is a right triangle, cos(DAC) = 0

  • AD^2 = 38.44 + 4 - 23.6

  • AD^2 = 19.24
  • AD ≈ 4.4 cm (rounded to one decimal place)

3. Apply the Law of Cosines to triangle ABD:

Now that we know AD = 4.4 cm, we can find BD using the Law of Cosines again:


  • BD^2 = AB^2 + AD^2 - 2 * AB * AD * cos(BAD)

  • BD^2 = 8.1^2 + 4.4^2 - 2 * 8.1 * 4.4 * cos(90°)

  • BD^2 = 65.61 + 19.36 - 72.24

  • BD^2 = 12.73
  • BD ≈ 3.6 cm (rounded to one decimal place)

Therefore, the length of side BD is approximately 3.6 cm and the length of side AD is approximately 4.4 cm.

User Fateh Khalsa
by
3.7k points
3 votes

Answer:

BD ≈ 2.8 units

Explanation:

By angle bisector theorem,

"If a segment bisects an angle of a triangle, then it divides the opposite side into the segments which are proportional to the other two sides"

Here, AD is the bisector of the ∠BAC,


(BD)/(CD)=(AB)/(AC)


(8.1)/(5.9)= (BD)/(2)

BD =
(2* 8.1)/(5.9)

BD = 2.75

BD ≈ 2.8 units

User Harutyun Abgaryan
by
4.7k points