Câu 1. Cho hình lăng trục đứng tam giác ABC.A' B'C...

Understanding triangular prism

I'm working through a problem involving a right triangular prism, specifically a right isosceles triangle at the base, calculating the volume given certain dimensions and angles.

Clarifying the triangle

I'm exploring the ambiguity of the triangle's description, noting a potential miscommunication or typo about whether ABC is a right isosceles triangle or a right triangle with AB as a leg.

Calculating prism height

I'm working through the prism's height, noting that AB' is a diagonal in the plane ABC to B' above B, and the angle gives insight into the height's calculation.

Determining prism height

Piecing together the height of a right prism, with AB' as a 45° diagonal from AB, I conclude that the distance B' to B (or A' to A or C' to C) is the height, h.

Mapping the projection

I’m noting that AB' projects onto plane ABC as AB, and I have a triangle ABB' with AB = 6 and angle(AB, AB') = 45°, which helps calculate B'B.

Taking a closer look

I’m exploring the right triangle formed by B' with A and the projection of B' onto plane ABC along AB. The angle between AB' and AB is 45°, aiding in calculating h.

Backtracking the angle

OK, let me see. Angle B'AB = 45°, but it doesn't imply a right angle since BB' is perpendicular to plane ABC. Systematic approaches are key.

Confirming geometric relationships

I’m mapping out that in a right prism, the height, h, is the perpendicular distance from B' to the plane ABC. The line AB' is a diagonal of a right triangle.

Determining the angle

I'm analyzing the angle between AB' and plane ABC, examining geometry and projection methods, specifically focusing on the right triangle formed by AB' and AB.

Analyzing perpendicularity

I'm noting that in triangle A-B-B', segment AB is not necessarily perpendicular to BB' unless both are parallel to their respective planes of orientation.

Determining triangle sides

I’m working out that AB' is the hypotenuse of a right triangle with legs AB (6) and BB' (h). The angle between AB' and plane ABC is the angle between AB' and AB.

Defining the angle

I'm thinking through how the angle between AB' and plane ABC is measured by the angle between AB' and its orthogonal projection onto the plane, which aligns with line AB.

Determining projections

I’m confirming that the projection of AB' onto plane ABC is line AB. The perpendicular drop from B' to the plane meets at point B, confirming the projection path.

Confirming the angle

I’m confirming that AB' forms a 45° angle with plane ABC, meaning angle B'AB is 45°. However, I’m beginning to rethink if triangle A-B-B' is the most suitable for this analysis.

Utilizing vector techniques

I’m mapping out a simpler approach using vectors to relate the angle between AB' and plane ABC to the lengths of segments AB and AB', considering their projections and ratios.

Connecting the dots

I'm gathering info on how the projection of a line onto a plane is related to cosine and the angle between the line and the plane, leading to a relationship with lengths.

Calculating height

Hm, that's interesting. I’m using Pythagoras to calculate the triangle's height. Then, I confirm the base shape to ensure the calculations match.

Calculating the base area

I’m working through the area of right isosceles triangle ABC using the leg lengths, then combining it with the prism height to find the volume.

Reasoned for 31 seconds