Question

drag each number to the correct location on the triangle. not all numbers will be used.
ivy is running errands for her mother. she bikes along straight paths to the supermarket, the bank, and then back home.

• ivy starts from her house at point a.
• first, she goes to the supermarket at point b.
• next, she goes to the bank at point c.
• last, she heads back to her house at point a.

find the distance between ivys house and the supermarket and the distance between the supermarket and the bank. each distance is rounded to the nearest meter.
854 m 2,717 m 1,014 m 3,635 m 1,523 m 2,288 m
note: figure not drawn to scale.

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(C\).
\(C = 180^{\circ}-(32^{\circ}+ 109^{\circ})=39^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
For side \(AB\) (distance from house \(A\) to supermarket \(B\)):
\(\frac{AB}{\sin C}=\frac{AC}{\sin B}\)
We know \(AC = 1523\) m, \(B = 109^{\circ}\), \(C=39^{\circ}\)
\(AB=\frac{1523\times\sin39^{\circ}}{\sin109^{\circ}}\)
\(\sin39^{\circ}\approx0.6293\), \(\sin109^{\circ}\approx0.9455\)
\(AB=\frac{1523\times0.6293}{0.9455}\approx1014\) m
For side \(BC\) (distance from supermarket \(B\) to bank \(C\)):
\(\frac{BC}{\sin A}=\frac{AC}{\sin B}\)
\(A = 32^{\circ}\), \(AC = 1523\) m, \(\sin32^{\circ}\approx0.5299\), \(\sin109^{\circ}\approx0.9455\)
\(BC=\frac{1523\times\sin32^{\circ}}{\sin109^{\circ}}=\frac{1523\times0.5299}{0.9455}\approx854\) m

Answer:

The distance between Ivy's house (\(A\)) and the supermarket (\(B\)) is \(1014\) m. The distance between the supermarket (\(B\)) and the bank (\(C\)) is \(854\) m.