Fill in the blanks in the proof below: Given: Right triangular prism with as base and height = Let the area of IJKLMN ∆IJK LI. be and the height of the prism is the altitude of to side ∆IJK “B” “h.” ∆IJK Prove: The volume for prism is equal to times V IJKLMN B h. First construct a right rectangular prism as follows: ABCDEFGH Construct rectangle ABCD with = and = = . Explain why the area of AB IJ AD KT rectangle = area of ABCD ∆IJK. Now construct right rectangular prism with the same height as prism ABCDEFGH This means that = _______ (an edge of prism IJKLM. BF IJKLM). Pass a plane parallel to the bases to intersect the two prisms. We can show that the formed are congruent to the bases, that is rectangle ____________ A’B’C’D’ and _______________. (See question 4 below for a formal proof.) ∆I’J’K’ But rectangle and have equal areas. Therefore rectangle and ABCD ∆IJK A’B’C’D’ have equal areas. ∆_______ also The plane that formed the cross sections could have been any plane parallel to the two bases. What can we now conclude by Postulate? Cavalieri’s In question 3c it was stated that the cross-sections of prism formed by planes parallel to the base are congruent to the base. show that this true in the case of the triangular Let’s prism. a. In the figure at the right || and || . Also and are all perpendicular to the base. ’ || , Explain why is a parallelogram. IJJ’I’ By the same reasoning and are also parallelograms. JKK’J’ KII’K’ b. Explain why = = and = IJ JK KI I’J’, J’K’ K’I’. c. Which congruence theorem can now be used prove that ∆IJK ∆I’J’K’? Prisms with Polygonal Bases Prism is shown at the right. ABCDEFGHIJ
Fill in the blanks in the proof below: Given: Right triangular prism IJKLMN with ∆IJK as base and height = LI. Let the area of ∆IJK be “B” and the height of the prism “h.” is the altitude of ∆IJK to side Prove: The volume V for prism IJKLMN is equal to B times h. First construct a right rectangular prism ABCDEFGH as follows: Construct rectangle ABCD with AB = IJ and AD = KT = . Explain why the area of rectangle ABCD = area of ∆IJK. Now construct right rectangular prism ABCDEFGH with the same height as prism IJKLM. This means that BF = _______ (an edge of prism IJKLM). Pass a plane parallel to the bases to intersect the two prisms. We can show that the formed are congruent to the bases, that is rectangle A’B’C’D’ ____________ and ∆I’J’K’ _______________. (See question 4 below for a formal proof.) But rectangle ABCD and ∆IJK have equal areas. Therefore rectangle A’B’C’D’ and ∆_______ also have equal areas. The plane that formed the cross sections could have been any plane parallel to the two bases. What can we now conclude by Cavalieri’s Postulate? In question 3c it was stated that the cross-sections of prism formed by planes parallel to the base are congruent to the base. Let’s show that this true in the case of the triangular prism. a. In the figure at the right ’ || , || and || . Also and are all perpendicular to the base. Explain why IJJ’I’ is a parallelogram. By the same reasoning JKK’J’ and KII’K’ are also parallelograms. b. Explain why IJ = I’J’, JK = J’K’ and KI = K’I’. c. Which congruence theorem can now be used prove that ∆IJK ∆I’J’K’? Prisms with Polygonal Bases Prism ABCDEFGHIJ is shown at the right.
Activity_6_3_3.docx