3.1 Distance, midpoint, volume and surface area questions for IB Maths AA

Find the coordinates of $\mathrm{B}$.

Find the exact value of $\mathrm{A}\mathrm{B}$.

Find the exact surface area of the sphere.

Find the value of $r$.

Find the total surface area of the paperweight.

Find $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$.

Find the exact area of triangle ABC.

The points ${\mathrm{A}}^{\mathrm{\prime }}$, ${\mathrm{B}}^{\mathrm{\prime }}$ and ${\mathrm{C}}^{\mathrm{\prime }}$ are the projections of A, B and C onto the $xy$-plane.

Use a vector product to find the area of triangle ${\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\mathrm{C}}^{\mathrm{\prime }}$.

Now consider any triangle ABC, with $\overrightarrow{\mathrm{A}\mathrm{B}}=\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right)$, $\overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right)$ and $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}{n}_1\\ n_2\\ n_3\end{array}\right)$.

Show that the area of the projection of triangle ABC onto the $xy$-plane is ${\displaystyle \frac{1}{2}}\mathrm{\mid }{n}_3\mathrm{\mid }$.

Write down, in terms of $n_1$, $n_2$ and $n_3$, the area of the projection of triangle ABC onto the $yz$-plane.

Write down, in terms of $n_1$, $n_2$ and $n_3$, the area of the projection of triangle ABC onto the $xz$-plane.

Let $T$ be the area of triangle ABC, and let $X$, $Y$ and $Z$ be the areas of its projections onto the $yz$-, $xz$- and $xy$-planes respectively.

Show that $T^2=X^2+Y^2+Z^2$.

Verify this result for the triangle in part (a).

The tetrahedron OPQR has vertices $\mathrm{O}(0,0,0)$, $\mathrm{P}(a,0,0)$, $\mathrm{Q}(0,b,0)$ and $\mathrm{R}(0,0,c)$, where $a,b,c>0$.

Find $\overrightarrow{\mathrm{P}\mathrm{Q}}\times \overrightarrow{\mathrm{P}\mathrm{R}}$.

Hence show that the areas of the four faces of the tetrahedron satisfy

Suggest why this result is sometimes described as a three-dimensional version of Pythagoras' theorem.

Consider the case $a=2$ and $b=3$.

Find the value of $c$ for which the area of face PQR is 7.

For the tetrahedron in part (k), let $\phi$ be the acute angle between the face PQR and the face OPQ.

Show that $\cos \phi ={\displaystyle \frac{\text{area of OPQ}}{\text{area of PQR}}}$, and hence find $\phi$.

Find the possible values of $k$.

For the integer value of $k$, find the coordinates of the midpoint of $[\mathrm{P}\mathrm{Q}]$.

Find the length of $[\mathrm{B}\mathrm{E}]$.

Find the angle between $[\mathrm{B}\mathrm{E}]$ and the base of the ramp.

Consider the case $a=2$, $b=3$ and $c=6$.

Find $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$.

Hence show that, in this case, the equation of $\mathrm{\Pi }$ is $3x+2y+z=6$.

Now consider the general case.

Show that $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}bc\\ ac\\ ab\end{array}\right)$.

Hence show that the equation of $\mathrm{\Pi }$ can be written as ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=1$.

Let $T$ be the area of triangle $\mathrm{A}\mathrm{B}\mathrm{C}$. Let $P$, $Q$ and $R$ be the areas of the right-angled triangles $\mathrm{O}\mathrm{B}\mathrm{C}$, $\mathrm{O}\mathrm{C}\mathrm{A}$ and $\mathrm{O}\mathrm{A}\mathrm{B}$ respectively.

For the case $a=2$, $b=3$ and $c=6$, find the value of $T$.

For the same case, find the values of $P$, $Q$ and $R$, and verify that $T^2=P^2+Q^2+R^2$.

Prove that $T^2=P^2+Q^2+R^2$ for all $a,b,c>0$.

Let $h$ be the perpendicular distance from $\mathrm{O}$ to $\mathrm{\Pi }$.

By considering the volume of the tetrahedron $\mathrm{O}\mathrm{A}\mathrm{B}\mathrm{C}$ in two different ways, show that

A plane meets the positive coordinate axes at $(k,0,0)$, $(0,k,0)$ and $(0,0,2k)$. The perpendicular distance from $\mathrm{O}$ to this plane is 4.

Find the value of $k$.

Let $\alpha$ be the acute angle between $\mathrm{\Pi }$ and the plane $x=0$, which contains triangle $\mathrm{O}\mathrm{B}\mathrm{C}$.

Show that $\cos \alpha ={\displaystyle \frac{P}{T}}$.

Let $\beta$ and $\gamma$ be the acute angles between $\mathrm{\Pi }$ and the planes $y=0$ and $z=0$ respectively.

Hence show that ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$.