Solve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.

01 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.

\begin{gathered}\text{B: }P=(8,2,11),\quad\alpha:z=0\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Identify the defining faces, cross-sections and generating elements of the solid.

Hint 2

B: Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Worked solution

Part A reasoning
Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$\lambda=1/2;\quad S_{\text{section}}/S_{\text{base}}=\lambda^2=1/4$
Area ratios are squares of linear ratios.
Part B reasoning
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$H=(8,2,0),\quad PH=11$
Distance to a plane is the perpendicular length, not distance to the origin.

A: The requested value is 0.25. B: The requested value is 11.

Checks and common pitfalls: Area ratios are squares of linear ratios. Distance to a plane is the perpendicular length, not distance to the origin.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

Solve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.

Introduction to solid geometry: mixed assessment

What you will be able to explain

Try it. Leave your reasoning visible.

Solve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.

Question sequence

Board plan

Anticipated thinking

Assessment checklist