In \(\triangle ABC\), \(D\) is a point on the side of \(BC\) such that \(AD = BD = CD\). If \(\angle BCA = 48^{\circ}\), then the size of \(\angle BAC\) is ____.
Answer (2)
To solve this problem, we need to understand the configuration of △ A BC with the given conditions.
Triangle Configuration :
We have △ A BC with point D on side BC such that A D = B D = C D . This means △ A B D , △ A C D , and △ A B D are isosceles triangles with respect to D .
Understanding ∠ BC A and Triangle Configuration :
Given ∠ BC A = 4 8 ∘ , this angle is part of △ A BC .
Since D is such that A D = B D = C D , A , B , C , and D form a cyclic quadrilateral (a quadrilateral where all vertices lie on a single circle).
Cyclic Quadrilateral Property :
In a cyclic quadrilateral, opposite angles are supplementary. Therefore, ∠ BC A + ∠ B A D = 18 0 ∘ .
Substitute the given ∠ BC A = 4 8 ∘ into the equation: ∠ B A C + 4 8 ∘ = 18 0 ∘
Solve for ∠ B A C :
Rearrange the equation to find ∠ B A C :
∠ B A C = 18 0 ∘ − 4 8 ∘ ∠ B A C = 13 2 ∘
Thus, the size of ∠ B A C is 13 2 ∘ .
In triangle ABC, with D on side BC such that AD = BD = CD, it can be concluded that angle BAC measures 132 degrees given that angle BCA is 48 degrees. The properties of cyclic quadrilaterals are used to derive this angle. Therefore, angle BAC is 132 degrees.
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