The figure below shows a triangular wooden frame abc. the side ad of the frame has rotted and needs to be replaced: what is the length of the wood that is needed to replace ad? 4.2 inches5.7 inches2.5 inches1.3 inches

AD = 4.2 in

Explanation:

Consider ΔABC, we have:

∠B = 90°

∠ACB = ∠ACD + ∠DCB = 15° + 30° = 45°

We know that the sum of the interior angles of a triangle is 180°

This means that:

∠BAC = 180° - 90° - 45° = 45°

Now, we have ∠BAC = ∠BCA which means that the triangle is isosceles and AB = BC = 10 in

Consider ΔBCD:

∠B = 90° and ∠DCB = 30°

We know that the sum of the interior angles of a triangle is 180°

This means that:

∠BDC = 180° - 90° - 30° = 60°

Now, a 30-60-90 triangle is a special type of triangle in which:

length of the side opposite to the 60° = the length of the hypotenuse

length of the side opposite to the 30° = the length of the hypotenuse

In ΔBDC, the side opposite to the 60° is 10 in

This means that:

Hypotenuse = CD =

BD = side opposite to the 30° = which is approximately 5.8 in

Finally:

We know that AB = 10 in and that BD = 5.8 in

Therefore:

AD = AB - BD = 10 - 5.8 = 4.2 in

Hope this helps :)

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