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Chapter 10: Problem 8

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=36 \mathrm{~cm}, \mathrm{BC}=36 \mathrm{~cm}, B=60^{\circ}\)

Short Answer

Expert verified
Question: Given an isosceles triangle with sides AB = 36 cm, BC = 36 cm, and angle B = 60°, find the remaining angles and side lengths. Answer: In this isosceles triangle, all sides are equal at 36 cm, and all angles are equal at 60°.

Step by step solution

01

Identify given information

Triangle \(\triangle ABC\) is given with sides \(\mathrm{AB}=36\,\mathrm{cm}\), \(\mathrm{BC}=36\,\mathrm{cm}\) and angle \(B=60^\circ\). This is an isosceles triangle where \(\mathrm{AB}=\mathrm{BC}\).
02

Find the third side using the Law of Cosines

Apply the Law of Cosines to find side \(\mathrm{AC}\). Since we have angle \(B\) and sides \(\mathrm{AB}\) and \(\mathrm{BC}\): \[AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos{B} \] Substitute the given values: \[AC^2 = 36^2 + 36^2 - 2 \cdot 36 \cdot 36 \cdot \cos{60^\circ} \] \[AC^2 = 1296 + 1296 - 2592 \cdot \frac{1}{2} \] \[AC^2 = 1296 \] \[AC = \sqrt{1296} = 36\, \mathrm{cm} \] So side \(\mathrm{AC} = 36\, \mathrm{cm}\).
03

Find the remaining angles

Since \(\mathrm{AB}=\mathrm{BC}=\mathrm{AC}\), the triangle is equilateral. Thus, all angles are equal: \[A=C=B=60^\circ\] So angles \(A\) and \(C\) are both \(60^\circ\).
04

Write the complete solution

We found that the triangle \(\triangle ABC\) is equilateral with all sides equal to \(36\, \mathrm{cm}\) and all angles equal to \(60^\circ\). So, the final solution is: \(\mathrm{AB}=\mathrm{BC}=\mathrm{AC}=36\, \mathrm{cm}\) \(A=B=C=60^\circ\)

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Most popular questions from this chapter

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AC}\) given \(\mathrm{BC}=12 \mathrm{~cm}\) and \(B=53^{\circ}\)

A ship travels for \(10 \mathrm{~km}\) on a bearing of \(30^{\circ} .\) It then follows a bearing of \(60^{\circ}\) for \(20 \mathrm{~km}\). Calculate the distance of the ship from the starting position.

A ship travels \(50 \mathrm{~km}\) from \(\mathrm{O}\) on a bearing of \(290^{\circ}\) to get to position A. From A it heads directly to B. Position B is \(90 \mathrm{~km}\) from \(\mathrm{O}\) on a bearing of \(190^{\circ}\). (a) Calculate the distance \(\mathrm{AB}\). (b) Calculate the bearing the ship must follow from \(\mathrm{A}\) to arrive directly at \(\mathrm{B}\).

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AC}\) given \(\mathrm{BC}=10 \mathrm{~cm}\) and \(A=40^{\circ}\).

For questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(A=37^{\circ}, B=47^{\circ}, \mathrm{AB}=17 \mathrm{~cm}\)
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