Please help with geometry homework (prisms)

[Lugo]

Hello,
Adding a picture with the problem, but it's in a foreign language, so i'll try to translate it the best i can.
We have a right-sided triangle at the base. Prism ABCA1B1C1 is split with an isosceles triangled plane A1BC.
Angle C is 90 degrees.
Dihedral corner angle is 45 degrees (angle which the split A1BC and the base ABC makes).
We know, that the new triangle A1BC is an isosceles triangle with side edges being 10cm. The volume of the prism is what i am looking for.

 

[Lugo]

This is how it would look like afterwards if my explanation wasn't clear enough

 

[blamocur]

I think your problem is assuming that [imath]A_1C = A_1 B[/imath], which is impossible since [imath]AB > AC[/imath]. Can you figure out which two sides of [imath]A_1BC[/imath] have equal lengths?

 

[The Highlander]

This is how it would look like afterwards if my explanation wasn't clear enoughView attachment 35728

I agree with @blamocur, the underlying assumption in your diagram is wrong.

I would suggest rather that it should look like this...

​

Now (since ΔA₁BC is isosceles) you have sufficient information to calculate the lengths of AC and AA₁ and, thence, the area of ΔABC and the volume of the prism.

I would further suggest that you work in surds rather than decimals (to preserve accuracy).

Pease post back to show us your solution. ?

 

[Dr.Peterson]

Hello,
Adding a picture with the problem, but it's in a foreign language, so i'll try to translate it the best i can.
We have a right-sided triangle at the base. Prism ABCA1B1C1 is split with an isosceles triangled plane A1BC.
Angle C is 90 degrees.
Dihedral corner angle is 45 degrees (angle which the split A1BC and the base ABC makes).
We know, that the new triangle A1BC is an isosceles triangle with side edges being 10cm. The volume of the prism is what i am looking for.





View attachment 35727

Here is the Google translation of the problem:

The base of a right prism ABCAB₁C₁ is a right triangle (ZC=90°). A prism intersects with a plane A₁BC. The dihedral angle formed by the cleavage plane and the base plane is 45°. For triangle A₁BC known to be an isosceles triangle whose side is 10 cm long. Calculate the volume of the prism.
(8 points)​

This makes a couple things clearer.

• It is a right prism, which means the vertical lines are all perpendicular to the base, ABC.
• The base ABC is a right triangle (that is, right-angled, not "right sided") with the right angle at C.
• The 45 degree angle is ACA1.

It leaves unclear which sides of A₁BC are equal, but since the angle at C would be a right angle, that must be the apex and it is A₁C and BC that have length 10.

This agrees with Highlander's picture, and this:


Now, as he said, find AC and AA1, then find the area of ABC, and the volume.

 

[The Highlander]

Here is the Google translation of the problem:

The base of a right prism ABCAB₁C₁ is a right triangle (ZC=90°). A prism intersects with a plane A₁BC. The dihedral angle formed by the cleavage plane and the base plane is 45°. For triangle A₁BC known to be an isosceles triangle whose side is 10 cm long. Calculate the volume of the prism.​

(8 points)​

This makes a couple things clearer.

• It is a right prism, which means the vertical lines are all perpendicular to the base, ABC.
• The base ABC is a right triangle (that is, right-angled, not "right sided") with the right angle at C.
• The 45 degree angle is ACA1.

It leaves unclear which sides of A₁BC are equal, but since the angle at C would be a right angle, that must be the apex and it is A₁C and BC that have length 10.

This agrees with Highlander's picture, and this:
View attachment 35733

Now, as he said, find AC and AA1, then find the area of ABC, and the volume.

(I was just leaving it for her/him to figure it out for themself that A₁C = BC. ??)