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Chapter 10: Problem 17

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{YZ}=15 \mathrm{~cm}, X=39^{\circ}\) and \(Y=75^{\circ}\).

Short Answer

Expert verified
Answer: In the triangle XYZ, the side lengths are YZ = 15 cm, XZ ≈ 18.65 cm, and XY ≈ 21.82 cm. The angle measures are X = 39°, Y = 75°, and Z = 66°.

Step by step solution

01

Find the third angle

We know that the sum of the angles in a triangle is always 180°. So, we can find the measure of angle Z by subtracting the measures of angles X and Y from 180°. Z = 180° - X - Y = 180° - 39° - 75° = 66° Now we know all the angles in the triangle: X = 39°, Y = 75°, and Z = 66°.
02

Use the sine rule to find side XZ

Since we know two angles and one side, we can use the sine rule to find another side length. Let's find the length of side XZ. The sine rule states that \(\frac{\text{side length}}{\sin(\text{opposite angle})}\) is the same for all sides in a triangle. So, for this triangle, we have: \(\frac{\text{YZ}}{\sin(X)}\) = \(\frac{\text{XZ}}{\sin(Y)}\) We know that YZ = 15 cm, X = 39°, and Y = 75°, so we can substitute these values into the sine rule equation: \(\frac{15}{\sin(39°)}\) = \(\frac{\text{XZ}}{\sin(75°)}\) Now, we can solve for XZ: XZ = \(\frac{15\cdot\sin(75°)}{\sin(39°)} \approx 18.65\,\text{cm}\)
03

Use the sine rule to find side XY

We can use the same sine rule to find the length of side XY: \(\frac{\text{YZ}}{\sin(X)}\) = \(\frac{\text{XY}}{\sin(Z)}\) We know that YZ = 15 cm, X = 39°, and Z = 66°, so we can substitute these values into the sine rule equation: \(\frac{15}{\sin(39°)}\) = \(\frac{\text{XY}}{\sin(66°)}\) Now, we can solve for XY: XY = \(\frac{15\cdot\sin(66°)}{\sin(39°)} \approx 21.82\,\text{cm}\)
04

Review the complete solution

Now we have found all the side lengths and angle measures for the triangle \(\Delta XYZ\): - Side YZ = 15 cm - Side XZ = 18.65 cm (approximately) - Side XY = 21.82 cm (approximately) - Angle X = 39° - Angle Y = 75° - Angle Z = 66° So, the triangle \(\Delta XYZ\) is completely solved.

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Most popular questions from this chapter

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

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{BC}=17 \mathrm{~cm}, \mathrm{AC}=27 \mathrm{~cm}, C=45^{\circ}\)

For questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(B=42^{\circ}, C=93^{\circ}, \mathrm{BC}=13 \mathrm{~cm}\)

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AB}\) given \(\mathrm{AC}=9 \mathrm{~cm}\) and \(\mathrm{BC}=15 \mathrm{~cm}\)

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(Z=46^{\circ}, \mathrm{YZ}=30 \mathrm{~m}\) and \(\mathrm{XY}=25 \mathrm{~m}\).
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