These 22 problems practice Common Core standards G-SRT.D.9, G-SRT.D.10 and G-SRT.D.11. Find triangle areas from two sides and an angle, apply the Law of Sines and the Law of Cosines, complete a short proof, and solve non-right triangles.
The last six problems are surveying and navigation word problems. Work from easiest to hardest, keep extra decimals until the final step, and use degree mode.
Problem 1: Area from two sides and an angle
Find the area of the triangle in square units.
Problem 2: Area, decimal answer
Two sides of a triangle measure 12 cm and 15 cm, and the angle between them is \(50^{\circ}\). Find the area to the nearest tenth of a square centimeter.
Problem 3: Area with an obtuse angle
A triangle has sides 7.5 in. and 10 in. with an included angle of \(115^{\circ}\). Find its area to the nearest tenth of a square inch.
Problem 4: Angle from the area
A triangle with area 60 square units has sides 10 and 15 on either side of an acute angle \(C\). Find \(m\angle C\) to the nearest tenth of a degree.
Problem 5: Law of Sines for a side
Find side \(b\) to the nearest tenth.
Problem 6: Third angle, then two sides
In triangle ABC, \(B=35^{\circ}\), \(C=80^{\circ}\) and \(c=24\). Find \(m\angle A\), then sides \(a\) and \(b\) to the nearest tenth.
Problem 7: Law of Sines for an angle
Find \(m\angle B\) to the nearest tenth of a degree. Side \(a\) is the longer side, so \(B\) is acute.
Problem 8: Law of Cosines, SAS
In triangle ABC, \(a=7\), \(b=12\) and \(C=60^{\circ}\). Find \(c\) to the nearest tenth.
Problem 9: Law of Cosines, obtuse angle
Find side \(a\) to the nearest tenth.
Problem 10: Law of Cosines, SSS
A triangle has sides 7, 9 and 12. Find the largest angle to the nearest tenth of a degree.
Problem 11: All three angles
Find all three angles of the triangle with sides 13, 14 and 15. Round each to the nearest tenth of a degree and check that they add to \(180^{\circ}\).
Problem 12: Choose the law
For each triangle, name the law (Sines or Cosines) you would use first to find an unknown side or angle, and explain why.
- \(a=9,\ b=12,\ C=47^{\circ}\)
- \(A=52^{\circ},\ B=71^{\circ},\ a=15\)
- \(a=7,\ b=9,\ c=12\)
- \(A=35^{\circ},\ a=7,\ b=10\)
Problem 13: Cosines and the Pythagorean theorem
Show that the Law of Cosines becomes the Pythagorean theorem when \(C=90^{\circ}\). Then verify with \(a=6\) and \(b=8\).
Problem 14: Proof of the area formula
Complete the proof that the area of triangle ABC is \(\frac{1}{2}ab\sin C\) by writing a reason for each step, (a) through (e). Here \(a=BC\), \(b=AC\) and \(\angle C\) is acute.
| Step | Statements | Reasons |
|---|---|---|
| 1 | Draw altitude \(\overline{AD}\) to \(\overline{BC}\), with \(AD=h\) | (a) |
| 2 | \(\text{Area}=\frac{1}{2}ah\) | (b) |
| 3 | \(\sin C=\frac{h}{b}\) | (c) |
| 4 | \(h=b\sin C\) | (d) |
| 5 | \(\text{Area}=\frac{1}{2}ab\sin C\) | (e) |
Problem 15: Area after the Law of Sines
In triangle ABC, \(A=40^{\circ}\), \(B=75^{\circ}\) and \(a=12\). Find the area of the triangle to the nearest tenth of a square unit.
Problem 16: Area of a parallelogram
A parallelogram has sides 9 and 14 and an angle of \(65^{\circ}\) between them. Find its area to the nearest tenth. (Hint for the figure: the dashed diagonal splits it into two congruent triangles.)
Problem 17: Distance across a pond
A surveyor at point C measures 420 ft to point A and 350 ft to point B on opposite sides of a pond. The angle \(\angle ACB\) is \(70^{\circ}\). How far apart are A and B? Round to the nearest foot.
Problem 18: Two boats leave a dock
Two boats leave the same dock at the same time on paths that form a \(70^{\circ}\) angle. After an hour, one boat is 18 mi from the dock and the other is 24 mi from it. How far apart are the boats? Round to the nearest tenth of a mile.
Problem 19: Sod for a triangular lot
A triangular lot has two sides of 120 ft and 85 ft that meet at a \(74^{\circ}\) angle. Find the area to the nearest square foot. Then find the cost of sod at $0.45 per square foot, using the rounded area.
Problem 20: Locating a fire
Fire towers A and B are 12 mi apart. The ranger at A sees a fire at an angle of \(52^{\circ}\) from line AB, and the ranger at B sees it at \(71^{\circ}\) from line BA. How far is the fire from each tower? Round to the nearest tenth of a mile.
Problem 21: Ship with a turn
A ship sails 120 mi due east, turns \(33^{\circ}\) toward the north, and then sails 80 mi. How far is the ship from its starting point? Round to the nearest tenth of a mile.
Problem 22: Garden angle and area
A triangular garden has sides 14 ft, 18 ft and 22 ft. Find the largest angle to the nearest tenth of a degree, then use it to find the area of the garden to the nearest tenth of a square foot.
See every chapter on the Geometry math page.