-
In the right-angled triangle \(\triangle ABC\), \(A\hat{C}B=90^\circ,~ A\hat{B}C=37^\circ25'\) and
\(a=BC=14~\mathrm{cm}\).
(a) Find the lengths of the other two sides \(b=AC\) and \(c=AB\).
(b) Calculate the perimeter \(P\) of the triangle.
Solutions:
(a) \(b\approx10.7~\mathrm{cm},~ c\approx17.6~\mathrm{cm}\);
(b) \(P=a+b+c\approx42.3~\mathrm{cm}\)
-
In the right-angled triangle \(\triangle ABC\), \(A\hat{C}B=90^\circ,~ A\hat{B}C=30^\circ\) and
\(a=BC=12~\mathrm{cm}\).
(a) Find the lengths of the other two sides \(b=AC\) and \(c=AB\).
(b) Calculate the perimeter \(P\) of the triangle.
Solutions:
(a) \(b=4\sqrt{3}~\mathrm{cm},~ c=8\sqrt{3}~\mathrm{cm}\);
(b) \(P=a+b+c=12+12\sqrt{3}~\mathrm{cm}\)
-
In the isosceles triangle \(\triangle ABC\), \(a=b=7~\mathrm{cm}\) and \(\alpha=B\hat{A}C=68^\circ\).
(a) Find the length of the side \(c\).
(b) Find the length of the height \(h_c\).
Solutions:
(a) \(c\approx5.24~\mathrm{cm}\);
(b) \(h_c\approx6.49~\mathrm{cm}\)
-
The isosceles triangle \(\triangle ABC\) \((a=b)\) has the base \(c=AB=24~\mathrm{cm}\)
and the height \(h_c=35~\mathrm{cm}\)
(a) Find the perimeter of this triangle.
(b) Calculate the angles of this triangle. Write them in degrees and minutes.
Solutions:
(a) \(P=24+37+37=98~\mathrm{cm}\);
(b) \(\alpha=\beta\approx71^\circ5',~ \gamma\approx37^\circ51'\)
-
The rectangle \(ABCD\) has the diagonal \(e=AC=20~\mathrm{cm}\).
The angle between this diagonal and the side \(a=AB\) is \(C\hat{A}B=27^\circ45'\).
(a) Find the sides of this rectangle.
(b) Calculate the acute angle between the diagonals.
Solutions:
(a) \(a\approx17.7~\mathrm{cm},~ b\approx9.31~\mathrm{cm}\);
(b) \(\varphi=55^\circ30'\)
-
The rectangle \(ABCD\) has the side \(a=AB=18~\mathrm{cm}\).
The angle between the diagonal \(d=AC\) and the side \(a\) is \(C\hat{A}B=30^\circ\).
Find the side \(b=BC\) and the diagonal \(d=AC\).
Write your results in the exact form.
Solutions:
\(b=6\sqrt{3}~\mathrm{cm},~ d=12\sqrt{3}~\mathrm{cm}\)
-
The diagonals of the rhombus \(ABCD\) have lengths \(e=AC=30~\mathrm{cm}\) and \(f=BD=16~\mathrm{cm}\).
(a) Find the side \(a=AB\).
(b) Calculate the angle \(\alpha=D\hat{A}B\).
Solutions:
(a) \(a=17~\mathrm{cm}\);
(b) \(\alpha\approx56^\circ9'\)
-
A regular pentagon \(ABCDE\) is inscribed in the circle with the radius \(r=10~\mathrm{cm}\).
Find the side \(a=AB\) of this pentagon.
Solutions:
\(a\approx11.8~\mathrm{cm}\)
-
A regular hexagon \(ABCDEF\) has the side \(a=7~\mathrm{cm}\).
Find the length of the diagonal \(d=AE\).
Solutions:
\(d=7\sqrt{3}~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(\alpha=C\hat{A}B=60^\circ,~ \beta=A\hat{B}C=45^\circ\)
and the height \(h_c=4\sqrt{3}~\mathrm{cm}\).
Find the lengths of the sides \(a,~ b\) and \(c\). Write the exact values.
Solutions:
\(a=4\sqrt{6}~\mathrm{cm},~ b=8~\mathrm{cm},~ c=(4+4\sqrt{3})~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(b=AC=6~\mathrm{cm},~ c=AB=10~\mathrm{cm}\) and \(\alpha=C\hat{A}B=70^\circ\).
The height \(h_c\) has the endpoints \(C\) and \(H\).
(a) Find the height \(h_c\).
(b) Find the distances \(AH\) and \(HB\).
(c) Hence, calculate the side \(a=BC\).
Solutions:
(a) \(h_c\approx5.64~\mathrm{cm}\);
(b) \(AH\approx2.05~\mathrm{cm},~ HB\approx7.95~\mathrm{cm}\);
(c) \(a\approx9.74~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(b=AC=5~\mathrm{cm},~ c=AB=8~\mathrm{cm}\) and \(\alpha=C\hat{A}B=60^\circ\).
Find the length of the side \(a\).
Solutions:
\(a=7~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(a=BC=13~\mathrm{cm},~ b=AC=19~\mathrm{cm}\) and \(\gamma=A\hat{C}B=64^\circ30'\).
Find the length of the side \(c\).
Solutions:
\(c\approx17.8~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(a=BC=9~\mathrm{cm},~ b=AC=14~\mathrm{cm}\) and \(c=AB=13~\mathrm{cm}\).
Calculate the angles \(\alpha,\beta\) and \(\gamma\). Round the results to the nearest minute.
Solutions:
\(\alpha\approx38^\circ43',~ \beta\approx76^\circ39',~ \gamma\approx64^\circ37'\)
-
In the triangle \(\triangle ABC\), \(a=BC=13~\mathrm{cm},~ b=AC=8~\mathrm{cm}\) and \(c=AB=7~\mathrm{cm}\).
Calculate the angles \(\alpha,\beta\) and \(\gamma\). Round the results to the nearest minute.
Solutions:
\(\alpha=120^\circ,~ \beta\approx32^\circ12',~ \gamma\approx27^\circ48'\)
-
In the triangle \(\triangle ABC\), \(a=BC=2~\mathrm{cm},~ b=AC=7~\mathrm{cm}\) and \(c=AB=3\sqrt{3}~\mathrm{cm}\).
Calculate the angle \(\beta\).
Solutions:
\(\beta=150^\circ\)
-
In the triangle \(\triangle ABC\), \(a=23~\mathrm{cm},~ c=17~\mathrm{cm}\) and \(\beta=107^\circ\).
(a) Find the side \(b\).
(b) Calculate the angles \(\alpha\) and \(\gamma\).
Solutions:
(a) \(b\approx32.4~\mathrm{cm}\);
(b) \(\alpha\approx42^\circ50',~ \gamma\approx30^\circ10'\)
-
In the triangle \(\triangle ABC\), \(\alpha=30^\circ,~ \beta=71^\circ\) and \(b=16~\mathrm{cm}\).
(a) Find the height \(h_c\).
(b) Calculate the side \(a\).
Solutions:
(a) \(h_c=8~\mathrm{cm}\);
(b) \(a\approx8.46~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(\alpha=57^\circ,~ \beta=71^\circ\) and \(c=15~\mathrm{cm}\).
(a) Find the angle \(\gamma\).
(b) Find the sides \(a\) and \(b\).
Solutions:
(a) \(\gamma=52^\circ\);
(b) \(a\approx16.0~\mathrm{cm},~ b\approx18.0~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(a=23~\mathrm{cm},~ b=39~\mathrm{cm}\) and \(\beta=83^\circ\).
(a) Find the angles \(\alpha\) and \(\gamma\).
(b) Find the side \(c\).
Solutions:
(a) \(\alpha\approx35^\circ50',~ \gamma\approx61^\circ10'\);
(b) \(c\approx34.4~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\), \(b=6~\mathrm{cm},~ c=8~\mathrm{cm}\) and \(\alpha=30^\circ\).
(a) Find the height \(h_c\).
(b) Find the area \(A\).
Solutions:
(a) \(h_c=3~\mathrm{cm}\);
(b) \(A=12~\mathrm{cm}^2\)
-
In the triangle \(\triangle ABC\), \(a=14~\mathrm{cm},~ c=11~\mathrm{cm}\) and \(\beta=77^\circ33'\).
(a) Find the side \(b\).
(b) Find the area \(A\).
Solutions:
(a) \(b\approx15.8~\mathrm{cm}\);
(b) \(A\approx75.2~\mathrm{cm}^2\)
-
In the triangle \(\triangle ABC\), \(a=5~\mathrm{cm},~ b=2~\mathrm{cm}\) and \(\gamma=58^\circ\).
Find the area of this triangle.
Solutions:
\(A\approx4.24~\mathrm{cm}^2\)
-
In the triangle \(\triangle ABC\), \(a=7~\mathrm{cm},~ b=8~\mathrm{cm}\) and \(c=5~\mathrm{cm}\).
(a) Find the angle \(\alpha=C\hat{A}B\).
(b) Find the area \(A\).
Solutions:
(a) \(\alpha=60^\circ\);
(b) \(A=10\sqrt{3}~\mathrm{cm}\approx17.3~\mathrm{cm}^2\)
-
The triangle \(\triangle ABC\) has sides: \(a=26~\mathrm{cm},~ b=15~\mathrm{cm}\) and \(c=37~\mathrm{cm}\).
Find the area of this triangle.
Solutions:
\(A=156~\mathrm{cm}^2\)
-
The triangle \(\triangle ABC\) has sides: \(a=14~\mathrm{cm},~ b=19~\mathrm{cm}\) and \(c=7~\mathrm{cm}\).
Find the area of this triangle.
Solutions:
\(A\approx39.5~\mathrm{cm}^2\)
-
The triangle \(\triangle ABC\) has sides: \(a=6~\mathrm{cm},~ b=8~\mathrm{cm}\) and \(c=17~\mathrm{cm}\).
Find the area of this triangle.
Solutions:
A triangle with these sides doesn't exist.
-
The triangle \(\triangle ABC\) has the sides: \(a=17~\mathrm{cm},~ b=10~\mathrm{cm}\) and \(c=21~\mathrm{cm}\).
(a) Find the area.
(b) Find the height \(h_c\).
Solutions:
(a) \(A=84~\mathrm{cm}^2\);
(b) \(h_c=8~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\): \(\alpha=39^\circ,~ \beta=68^\circ\) and \(c=14~\mathrm{cm}\).
(a) Find the angle \(\gamma\).
(b) Find the sides \(a\) and \(b\).
(c) Find the perimeter and area.
Solutions:
(a) \(\gamma=73^\circ\);
(b) \(a\approx9.21~\mathrm{cm},~ b\approx13.6~\mathrm{cm}\);
(c) \(P\approx36.8~\mathrm{cm},~ A\approx59.8~\mathrm{cm}^2\)
-
In the parallelogram \(ABCD\): \(a=AB=15~\mathrm{cm},~ b=BC=7~\mathrm{cm}\) and \(\alpha=B\hat{A}D=60^\circ\).
(a) Find the diagonal \(f=BD\).
(b) Find the area of this parallelogram.
Solutions:
(a) \(f=13~\mathrm{cm}\);
(b) \(A\approx90.9~\mathrm{cm}^2\)
-
The rhombus \(ABCD\) has the side \(a=7~\mathrm{cm}\) and the angle \(\alpha=69^\circ\).
(a) Find the diagonal \(f=BD\).
(b) Find the height \(h\).
(c) Find the area of this rhombus.
Solutions:
(a) \(f\approx7.93~\mathrm{cm}\);
(b) \(h\approx6.54~\mathrm{cm}\);
(c) \(A\approx45.7~\mathrm{cm}^2\)
-
A regular pentagon is inscribed in the circle with the radius \(r=8~\mathrm{cm}\).
(a) Find the side of this pentagon.
(b) Find the diagonal.
(c) Find the area.
Solutions:
(a) \(a\approx9.40~\mathrm{cm}\);
(b) \(d\approx15.2~\mathrm{cm}\);
(c) \(A\approx152~\mathrm{cm}^2\)
-
A regular octagon \(ABCDEFGH\) has the side \(a=10~\mathrm{cm}\).
(a) Find the diagonal \(AC\).
(b) Find the inradius \(r\) (radius of the inscribed circle).
(c) Find the area of this octagon.
Solutions:
(a) \(AC\approx18.5~\mathrm{cm}\);
(b) \(r\approx12.1~\mathrm{cm}\);
(c) \(A\approx483~\mathrm{cm}^2\)
-
A regular nonagon has the side \(a=6~\mathrm{cm}\).
(a) Find the circumradius \(R\) (radius of the circumscribed circle).
(b) Find the length of the longest diagonal.
(c) Find the area of this nonagon.
Solutions:
(a) \(R\approx8.77~\mathrm{cm}\);
(b) \(d\approx17.3~\mathrm{cm}\);
(c) \(A\approx223~\mathrm{cm}^2\)
-
In the triangle \(\triangle ABC\): \(a=2~\mathrm{cm},~ b=7~\mathrm{cm}\) and \(c=3\sqrt{3}~\mathrm{cm}\).
(a) Find the angle \(\alpha\).
(b) Find the angle \(\beta\). Try calculating \(\beta\) using the cosine rule and using the sine rule.
Solutions:
(a) \(\alpha\approx8^\circ13'\);
(b) \(\beta=150^\circ\) (and not \(30^\circ\))
-
In the triangle \(\triangle ABC\): \(a=19~\mathrm{cm},~ b=21~\mathrm{cm}\) and \(\alpha=60^\circ\).
(a) Find the angle \(\beta\). Write down both possible values of \(\beta\).
(b) Find the side \(c\). Write down both possible values of \(c\).
Solutions:
(a) \(\beta_1\approx73^\circ10',~ \beta_2\approx106^\circ50'\);
(b) \(c_1=16~\mathrm{cm},~ c_2=5~\mathrm{cm}\)
-
In the triangle \(\triangle ABC\): \(a=7~\mathrm{cm},~ b=13~\mathrm{cm}\) and \(\alpha=105^\circ\).
Find the angle \(\beta\) and the side \(c\).
Solutions:
Such a triangle can't exist.
-
In the triangle \(\triangle ABC\): \(a=7~\mathrm{cm},~ b=13~\mathrm{cm}\) and \(\alpha=68^\circ\).
Find the angle \(\beta\) and the side \(c\).
Solutions:
Such a triangle can't exist.
-
In the triangle \(\triangle ABC\): \(a=7~\mathrm{cm},~ b=13~\mathrm{cm}\) and \(\alpha=29^\circ\).
Find the angle \(\beta\) and the side \(c\).
Solutions:
Two solutions:
(1) \(\beta_1\approx64^\circ12',~ c_1\approx14.4~\mathrm{cm}\);
(2) \(\beta_2\approx115^\circ48',~ c_2\approx8.32~\mathrm{cm}\)
-
The triangle \(\triangle ABC\) has the sides \(a=5~\mathrm{cm},~ b=8~\mathrm{cm}\) and the area \(A=8~\mathrm{cm}^2\).
Find the side \(c\).
Solutions:
Two solutions:
\(c_1\approx3.96~\mathrm{cm},~ c_2\approx12.7~\mathrm{cm}\)
-
The circle has the radius \(r=5~\mathrm{cm}\).
Find the area and perimeter of this circle.
Solutions:
\(A=25\pi~\mathrm{cm}^2\approx78.5~\mathrm{cm}^2,~ P=10\pi~\mathrm{cm}\approx31.4~\mathrm{cm}\)
-
The circle has the circumference \(100~\mathrm{cm}\).
Find the area of this circle.
Solutions:
\(r\approx15.9~\mathrm{cm},~ A\approx796~\mathrm{cm}^2\)
-
Write the following angles in radians:
(a) \(30^\circ\)
(b) \(45^\circ\)
(c) \(120^\circ\)
(d) \(135^\circ\)
Solutions:
(a) \(30^\circ=\frac{\pi}{6}\);
(b) \(45^\circ=\frac{\pi}{4}\);
(c) \(120^\circ=\frac{2\pi}{3}\);
(d) \(135^\circ=\frac{3\pi}{4}\)
-
Convert the following angles to degrees:
(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{2}\)
(c) \(\frac{5\pi}{6}\)
(d) \(\frac{5\pi}{4}\)
Solutions:
(a) \(\frac{\pi}{3}=60^\circ\);
(b) \(\frac{\pi}{2}=90^\circ\);
(c) \(\frac{5\pi}{6}=150^\circ\);
(d) \(\frac{5\pi}{4}=225^\circ\)
-
The circle has the radius \(r=12~\mathrm{cm}\).
Find the length of the arc which subtends the angle \(\theta=40^\circ\) at the centre of this circle.
Solutions:
\(L=\frac{8}{3}\pi~\mathrm{cm}\approx8.38~\mathrm{cm}\)
-
The circle has the radius \(r=9~\mathrm{cm}\).
Find the length of the arc which subtends the angle \(\theta=2~\mathrm{radians}\).
Solutions:
\(L=18~\mathrm{cm}\)
-
Find the area of the sector of the circle with the radius \(r=18~\mathrm{cm}\)
and the central angle \(\theta=10^\circ\).
Solutions:
\(A=9\pi~\mathrm{cm}^2\approx28.3~\mathrm{cm}^2\)
-
Given the radius \(r=6~\mathrm{cm}\) and central angle \(\theta=\frac{1}{8}\pi\),
(a) find the area of the circular sector,
(b) find the length of the arc.
Solutions:
(a) \(A\approx7.07~\mathrm{cm}^2\);
(b) \(L\approx2.36~\mathrm{cm}\)
-
Given the radius \(r=8~\mathrm{cm}\) and central angle \(\theta=90^\circ\),
(a) find the area of the circular sector,
(b) find the area of the corresponding circular segment.
Solutions:
(a) \(A_1\approx50.3~\mathrm{cm}^2\);
(b) \(A_2\approx18.3~\mathrm{cm}^2\)
-
Find the area of the circular segment of the radius \(r=14~\mathrm{cm}\) and
central angle \(\theta=1\).
Solutions:
\(A\approx15.5~\mathrm{cm}^2\)
