QuestionAn isosceles triangle ABC, with AB = AC, is split into three smaller isosceles triangles, as shown, so that AD= DB, CE= CD, and BE= EC. (Note that the diagram is not drawn to scale.) What is the size, in degrees, of angle BAC? |
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| A: 24 | B: 28 | C: 30 | D: 35 | E: 36 |
LLM Response: Let's approach this step-by-step: 1) In an isosceles triangle, the angles opposite the equal sides are equal. So in triangle ABC, angles B and C are equal. 2) Since the sum of angles in a triangle is 180°, we can say: BAC + B + C = 180° BAC + 2B = 180° (since B = C) 3) Now, let's look at the smaller triangles: - In triangle ABD, AD = DB, so angles BAD and ABD are equal. Let's call each of these x°. - In triangle ACD, AC = CD, so angles CAD and ACD are equal. Let's call each of these y°. - In triangle BCE, BE = EC, so angles CBE and BCE are equal. Let's call each of these z°. 4) Now, angle B = x° + z° (external angle of triangle ABD) And angle C = y° + z° (external angle of triangle ACD) 5) Since B = C, we can say x° + z° = y° + z°, which means x° = y° 6) Now, angle BAC = x° + y° = 2x° (since x° = y°) 7) Remember from step 2: BAC + 2B = 180° We can now write this as: 2x° + 2(x° + z°) = 180° 2x° + 2x° + 2z° = 180° 4x° + 2z° = 180° 8) In triangle BCE, 2z° + 60° = 180° (sum of angles in a triangle) So, z° = 60° 9) Substituting this in the equation from step 7: 4x° + 2(60°) = 180° 4x° + 120° = 180° 4x° = 60° x° = 15° 10) Remember, BAC = 2x° = 2(15°) = 30° Therefore, the size of angle BAC is 30°. C3: 30
Final Prediction: Puzzle #89: Solve this question with explanation of the intermediate steps. Your response must end with one of the selected answer options from A1, B2, C3, D4, or E5. An isosceles triangle ABC, with AB = AC, is split into three smaller isosceles triangles, as shown, so that AD= DB, CE= CD, and BE= EC. (Note that the diagram is not drawn to scale.) What is the size, in degrees, of angle BAC? Options: A1: 24, B2: 28, C3: 30, D4: 35, E5: 36. Predicted Answer = C Correct Answer = E