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14 problems match “triangle area”. Clear

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1. Area of a triangle from vectors ●○○○○ 10 ptsLet $\mathbf{a} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$. Find the area of the triangle with vertices $\mathbf{0}$, $\mathbf{a}$, and $\mathbf{b}$.
2. Right triangle with given circumradius and altitude ●○○○○ 10 ptsA right-angled triangle has circumradius $3$ cm. The altitude drawn to its hypotenuse is $2$ cm. Find the area of the triangle in square centimetres.
3. Area of a triangle in polar coordinates ●○○○○ 10 ptsIn the polar coordinate system, points $A\left(2, \frac{\pi}{6}\right)$ and $B\left(4, \frac{\pi}{3}\right)$ are given, and $O$ is the pole. Find the area $S$ of $\triangle AOB$.
4. Shortest altitude of a triangle ●●○○○ 20 ptsThe sides of a triangle have lengths $15$, $20$, and $25$. Find the length of the shortest altitude.
5. Triangle area from a dot product ●●○○○ 20 ptsIn triangle $ABC$, $AB=1$, $BC=3$, and $\overrightarrow{AB}\cdot\overrightarrow{BC}=-1$. Find the area of triangle $ABC$. Round to two decimal places.
6. Area of triangle from transformed vertices ●●○○○ 20 ptsThe triangle with vertices $\vec{A},$ $\vec{B},$ and $\vec{C}$ has area 12. Find the area of the triangle whose vertices are $-\vec{A} + \vec{B} + \vec{C},$ $\vec{A} - \vec{B} + \vec{C},$ and $\vec{A} + \vec{B} - \vec{C}.$
7. Volume of tetrahedron from dihedral angle ●●○○○ 20 ptsFaces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
8. Parallelogram area from sides and diagonal ●●○○○ 20 ptsTwo adjacent sides of a parallelogram are $50$ cm and $40$ cm, and one diagonal is $30$ cm. Find the area of the parallelogram in square centimetres.
9. Triangle sides from heights ●●○○○ 20 ptsIn triangle $ABC$, the height to side $a$ is $6$ cm and the height to side $b$ is $9$ cm. Side $a$ is $4$ cm longer than side $b$. Calculate the side lengths $a$ and $b$ (in cm).
10. Area of a 30-60-90 triangle ●●○○○ 20 ptsA right triangle has angles of $30^\circ$ and $60^\circ$. The altitude from the right-angle vertex to the hypotenuse is $3$ units. Find the area of the triangle in square units. Round your answer to two decimal places.

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