1
In right triangle \(ABC\), \(\angle C\) is a right angle. The side lengths are \(AC=8\), \(BC=15\), and \(AB=17\). What is the value of \(\tan A\)?
Note: The figure is not drawn to scale.
Right triangles and trigonometry: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
Triangles \(\triangle PQR\) and \(\triangle XYZ\) are similar. Angle \(Q\) and angle \(Y\) are right angles, and angle \(P\) corresponds to angle \(X\). If \(\cos P=\frac{5}{8}\), what is the value of \(\cos X\)?
3
Triangle \(JKL\) is similar to triangle \(PQR\). Angles \(K\) and \(Q\) are right angles, and angle \(L\) corresponds to angle \(R\). If \(\cos L=\frac{5}{8}\), what is the value of \(\cos R\)?
4
Triangle \(RST\) is a right triangle with a right angle at \(S\). The length of \(RT\) is 21, and the length of \(RS\) is 9.
What is the value of \(\cos R\)?
Note: The figure is not drawn to scale.
5
Triangle ABC is a right triangle with a right angle at C. The lengths of sides AC and BC are both 18. If the measure of angle A is \(y^ irc\), what is the value of \(y\)?
The figure may not be drawn to scale.
6
Triangle ABC is a right triangle with a right angle at vertex C. The lengths of sides AC and BC are both 18. Angle A is labeled \(y\). What is the value of \(y\)?
Note: The figure may not be drawn to scale.
7
The figure shows right triangle \(ABC\), where \(\angle C\) is a right angle. Side \(AC\) has length 11, side \(AB\) has length 19, and side \(BC\) has length \(t\).
Which expression represents the value of \(t\)?
Note: The figure is not drawn to scale.
8
The figure shows right triangle ABC with a right angle at C. Side AC has length 9, side AB has length 15, and side BC has length m. The figure is not drawn to scale.
Which expression represents the value of m?
9
The acute angles \(P\) and \(Q\) satisfy the equation \(\cos(P)=\sin(Q)\). If the measure of angle \(P\) is \(4x+13\) degrees and the measure of angle \(Q\) is \(3x+7\) degrees, what is the value of \(x\)?
10
Right triangles \(\triangle JKL\) and \(\triangle RST\) are similar, with \(\angle K\) corresponding to \(\angle S\). The length of each side in \(\triangle RST\) is \(2.5\) times the length of the corresponding side in \(\triangle JKL\). If \(\tan(K)=\frac{20}{21}\), what is the value of \(\sin(S)\)?
11
Right triangles \(\triangle JKL\) and \(\triangle RST\) are similar, with \(\angle K\) corresponding to \(\angle S\). The length of each side in \(\triangle RST\) is \(2.5\) times the length of the corresponding side in \(\triangle JKL\). If \(\tan K=\frac{20}{21}\), what is the value of \(\sin S\)?
12
A rectangle has a width of \(9\) units and a diagonal of length \(3\sqrt{34}\) units. What is the length, in units, of the longer side of the rectangle?
Answer key and explanations
1. Answer
\(\frac{15}{8}\)
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
For \(\angle A\), the opposite side is \(BC=15\), and the adjacent side is \(AC=8\).
So, \(\tan A=\frac{15}{8}\).
2. Answer
\(\frac{5}{8}\)
Since the triangles are similar, corresponding angles have equal trigonometric ratios. Angle \(P\) corresponds to angle \(X\), so \(\cos X=\cos P\). Because \(\cos P=\frac{5}{8}\), it follows that \(\cos X=\frac{5}{8}\).
3. Answer
\(\frac{5}{8}\)
Because triangles \(JKL\) and \(PQR\) are similar, corresponding angles are congruent. Since angle \(L\) corresponds to angle \(R\), the cosine ratios for those angles are equal. Therefore, \(\cos R=\cos L=\frac{5}{8}\).
4. Answer
\(\frac{3}{7}\)
Because the right angle is at \(S\), side \(RT\) is the hypotenuse. For angle \(R\), side \(RS\) is the adjacent side.
Cosine is defined as:
\(\cos R=\frac{\text{adjacent}}{\text{hypotenuse}}\)
Substitute the given side lengths:
\(\cos R=\frac{9}{21}=\frac{3}{7}\)
Therefore, the correct answer is \(\frac{3}{7}\).
5. Answer
45
Since sides AC and BC have the same length, triangle ABC is an isosceles triangle. Because angle C is a right angle, the other two angles must add to \(90^ irc\). In an isosceles triangle, the angles opposite the equal sides are equal, so angles A and B are congruent.
Therefore, each acute angle measures \(\frac{90}{2}=45\) degrees. So \(y=45\).
6. Answer
45
Since sides AC and BC have the same length, triangle ABC is an isosceles triangle. Because angle C is a right angle, the other two angles must add to \(90^\circ\). In an isosceles triangle, the two acute angles are equal, so each acute angle measures \(\frac{90}{2}=45\). Therefore, \(y=45\).
7. Answer
\(\sqrt{19^{2}-11^{2}}\)
Because \(\angle C\) is a right angle, side \(AB\) is the hypotenuse. By the Pythagorean theorem,
\(t^{2}+11^{2}=19^{2}\)
Subtract \(11^{2}\) from both sides:
\(t^{2}=19^{2}-11^{2}\)
Since a side length must be positive, take the principal square root:
\(t=\sqrt{19^{2}-11^{2}}\)
Therefore, the correct answer is choice C.
8. Answer
\(\sqrt{15^{2}-9^{2}}\)
Because AB is opposite the right angle, it is the hypotenuse. Apply the Pythagorean theorem:
\(m^{2}+9^{2}=15^{2}\)
Subtract \(9^{2}\) from both sides:
\(m^{2}=15^{2}-9^{2}\)
Since side lengths are positive, take the positive square root:
\(m=\sqrt{15^{2}-9^{2}}\)
Therefore, the correct answer is \(\sqrt{15^{2}-9^{2}}\).
9. Answer
10
For acute angles, the equation \(\cos(P)=\sin(Q)\) means that \(P\) and \(Q\) are complementary angles. Therefore, their measures add to \(90^\circ\).
Set up the equation:
\((4x+13)+(3x+7)=90\)
Combine like terms:
\(7x+20=90\)
Subtract 20 from both sides:
\(7x=70\)
Divide by 7:
\(x=10\)
Therefore, the correct answer is 10.
10. Answer
20/29 / 0.689655
Because the triangles are similar and \(\angle K\) corresponds to \(\angle S\), the angles have the same trigonometric ratios. Therefore, \(\tan(S)=\tan(K)=\frac{20}{21}\).
Tangent is opposite over adjacent, so the legs of a right triangle can be represented as 20 and 21. Using the Pythagorean theorem:
\(20^2+21^2=400+441=841\)
\(\sqrt{841}=29\)
So the hypotenuse is 29. Since sine is opposite over hypotenuse,
\(\sin(S)=\frac{20}{29}\).
The scale factor of 2.5 is not needed to solve the problem.
11. Answer
20/29 / 0.689655
Because the triangles are similar and \(\angle K\) corresponds to \(\angle S\), the angles are congruent, so \(\tan S=\tan K=\frac{20}{21}\).
A tangent ratio of \(\frac{20}{21}\) means the opposite and adjacent legs can be represented by 20 and 21. Using the Pythagorean theorem:
\(20^2+21^2=400+441=841\)
\(\sqrt{841}=29\), so the hypotenuse is 29.
Therefore, \(\sin S=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{20}{29}\).
12. Answer
15
The diagonal of a rectangle forms the hypotenuse of a right triangle whose legs are the side lengths of the rectangle. Let \(x\) be the longer side.
Using the Pythagorean theorem:
\(x^2+9^2=(3\sqrt{34})^2\)
Square each value:
\(x^2+81=9\cdot 34\)
\(x^2+81=306\)
Subtract 81 from both sides:
\(x^2=225\)
\(x=15\)
Since a side length must be positive, the longer side is \(15\) units.