Exercises: Theorems on triangles, parallelograms and tangents

Can a triangle have sides of lengths 3 cm, 4 cm and 8 cm? Use the triangle inequality to explain.

A triangle has base 10 cm and perpendicular height 6 cm. Calculate its area.

A circle has centre $O$ and a tangent touching it at point $T$. The radius is $5$ cm. State the angle between $OT$ and the tangent, and find the length of the tangent from a point $P$ with $|OP| = 13$ cm.

In triangle $ABC$, $|\angle A| = 70^\circ$, $|\angle B| = 50^\circ$.

1. a) Find $|\angle C|$.
2. b) Name the longest and the shortest side, giving the reason.

Two sides of a triangle are 7 cm and 12 cm long.

1. a) Between which two values must the third side $x$ lie?
2. b) If the triangle is isosceles, find the possible values of $x$.

A parallelogram $ABCD$ has $|AB| = 9$ cm, and the perpendicular distance between $AB$ and $DC$ is 5 cm.

1. a) Calculate the area of the parallelogram.
2. b) The diagonal $AC$ divides the parallelogram into two triangles. State the area of each and give a reason.

In triangle $PQR$, $M$ is the midpoint of $[QR]$. Prove that triangles $PQM$ and $PMR$ have equal areas. Hence, if the area of triangle $PQR$ is $48\text{ cm}^2$, state the area of $PQM$.

Two tangents from an external point $P$ touch a circle with centre $O$ at $A$ and $B$. The radius is 6 cm and $|OP| = 10$ cm.

1. a) Find $|PA|$ and explain why $|PA| = |PB|$.
2. b) Find the area of the quadrilateral $OAPB$.
3. c) Find $|\angle APB|$ to the nearest degree.

In triangle $ABC$, the point $D$ lies on $[BC]$. Triangles $ABC$ and $ABD$ share the vertex $A$.

1. a) If $|BD| : |DC| = 2 : 3$, show that the area of $ABD$ is $\frac{2}{5}$ of the area of $ABC$.
2. b) The parallelogram $ABEC$ is completed with $E$ so that $[AE]$ and $[BC]$ bisect each other. Show that the area of the parallelogram is twice the area of triangle $ABC$.