Find the area of the triangle with sides a, b and included angle γ.
a = 7, b = 4, γ = 45°
Answer: 7 √2
10th Grade Similarity, Right Triangles, and Trigonometry
Imaginesis: printable 10th Grade Similarity, Right Triangles, and Trigonometry worksheets for the United States (Common Core) — Laws of Sines and Cosines, Find Sides and Angles in Right Triangles, Areas and Perimeters of Similar Figures: word problems with hints, answer keys and step-by-step solutions as PDF, also as a test in groups A–D.
Sample word problems
In a right triangle the hypotenuse c and the acute angle α are given. Find the leg adjacent to α.
c = 12, α = 30°
Answer: 6 √3
Figure F₂ is similar to F₁ with the given scale factor. The area of F₁ is given. Find the area of F₂.
k = 2/4, P₁ = 45
Answer: 45/4
Skills in this topic
Laws of Sines and Cosines
Your child learns to use the Law of Sines, the Law of Cosines, and the sine area formula to find unknown sides, angles, and areas of triangles.
HSG-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.HSG-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unkno wn measurements in right and non-right triangles (e.g., surveying problems, resultant forces).HSG-SRT.D.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
At this stage, your child works with the Law of Sines and the Law of Cosines to find missing measurements in right and non-right triangles, and uses the area formula A = 1/2 ab sin(C). The curriculum points for this skill do not state a specific number range, so this page does not give one. The points also ask for the derivation and proof of these formulas, which is a further step beyond the printed practice.
A common error is squaring the wrong side in the Law of Cosines. The formula has one squared side on the left and a squared term plus a cross-product on the right, and it is easy to mix up which side is opposite the angle being used. Another frequent slip is getting the sign of the cosine term wrong, especially when the angle is obtuse. These errors happen because the parts of the formula look alike, so checking which side is opposite which angle before calculating helps.
Tasks for this skill are printed as worksheets and include:
- Law of cosines, including finding the missing side
- Law of cosines: choose the squared side
- Law of cosines: insert the sign
- Law of cosines: true or false
- Law of cosines: find the mistake
- Law of sines
- Area of a triangle with sine
- Triangle area with a sine: choose the result
Create a worksheet for this skill
Drill sheet (numbers only)
Task types: Area of a triangle with sine · Triangle area with a sine: choose the result · Law of cosines · Law of cosines: choose the squared side · Law of cosines: find the mistake · Law of cosines: insert the sign · Law of cosines: true or false · Law of cosines outdoors: the distance between two places · Law of sines
Find Sides and Angles in Right Triangles
Your child learns to use sine, cosine, and tangent to find a missing side or ratio in a right triangle, and to use the Pythagorean Theorem alongside those ratios.
HSG-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★
At this stage, your child works with right triangles and the three trigonometric ratios: sine, cosine, and tangent. The curriculum asks students to see that the ratio of two sides in a right triangle depends only on the angle, which is why these ratios can be defined for any acute angle. It also asks students to use these ratios and the Pythagorean Theorem to solve right triangles. The printed tasks in this area focus on identifying and using the ratios: picking the correct value for sine, cosine, or tangent, and identifying which side is opposite or adjacent to a given angle.
A common mistake is mixing up the sides. Sides are labeled relative to the angle in question, so the leg opposite one angle is the adjacent leg for the other acute angle. A child who reads the sides from the wrong angle, or who swaps sine and cosine, will get a ratio that looks reasonable but is wrong. Checking which side is opposite and which is adjacent before writing any ratio helps prevent this.
Tasks in this area include identifying the leg opposite or adjacent to an angle, choosing the correct value for sine or tangent in a given triangle, deciding whether a cosine statement is true or false, and finding the mistake in a worked cosine solution. These tasks ask your child to read the triangle carefully and check each ratio against the sides, which builds the habit needed for solving triangles later.
Create a worksheet for this skill
Drill sheet (numbers only)
Task types: Cosine in a right triangle: find the mistake · Cosine in a right triangle: true or false · Leg adjacent to an angle · Leg opposite an angle · Sine in a right triangle: choose the value · Tangent in a right triangle: choose the value · Trigonometry outdoors: the height at a 30° angle
Areas and Perimeters of Similar Figures
Your child learns how enlarging or shrinking a figure by a scale factor changes its perimeter and area, and uses that to find missing measures.
8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
The curriculum points behind this skill are written for grade 8 and high school, and they do not state a number range. In grade 10, the printed tasks ask your child to work with scaled copies of figures. Your child should be able to say whether two figures are similar, find the scale factor between them, and use it to find a missing side in similar triangles or the area of a scaled copy.
A common mistake is treating area like length. If a shape's sides are doubled, its perimeter doubles, but its area becomes four times as large, not two times. Students often multiply the area by the scale factor once instead of squaring it, which gives an answer that looks reasonable but is wrong.
Practice tasks on this skill include:
- Similar figures: the area of a scaled copy, where your child finds the area of an enlarged or reduced figure.
- Area of a similar figure, where the given area is used to find the area of a similar figure.
- Similar triangles: the missing side, where the scale factor is used to find an unknown side length.
- Similar figures: choose the area and true or false, which check whether your child understands how the scale factor affects area.
- Similar figures: find the mistake, where your child looks at a worked answer and finds the error, often the mistake of scaling area by the factor rather than its square.